Marie-Françoise Bidaut-Véron
email: veronmf@univ-tours.fr

Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term

Keywords: Liouville property, Bernstein method, Keller-Osserman estimates. MSC2010: 35J92

published or not. The documents may come

Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term

 [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]The case p < 0 24 [START_REF] Abdelhamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF]

Introduction

In this paper we study local and global properties of positive solutions of the equation -div |∇u| m-2 ∇u := -∆ m u = u p |∇u| q , (1.1) in R N , (N ≥ 1, 1 < m < N and p > 0) in the supercritical case

q ≥ m. (1.2)
We are concerned by the Liouville property in R N , which is wether all the positive C 1 solutions are constant. We also study the asymptotic behaviour of any solution of (1.1) near a singularity in the punctured ball B r 0 \ {0}, in R N \ {0} or in an exterior domain R N \B r 0 .

In the case q = 0, equation (1.1) reduces to the classical Lane-Emden-Fowler equation

-∆ m u = u p , (1.3)
which has already been the subject of countless publications. One of the questions solved is that the Liouville property holds if and only if p < p * m := N (m -1) + m N -m .

Note that p * m is the Sobolev exponent. Since it is impossible to quote all the articles on the subject, we only mention here the pioneering works and references therein. Gidas and Spruck [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] first showed the nonexistence of positive solutions in R N for m = 2 and p < p * 2 . They combine the Bernstein technique applied in the equation satisfed by the gradient of a suitable power of u, with delicate integral estimates ensuring the Harnack inequality, see also [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. Then the complete behaviour up to the case p = p * 2 was obtained by moving plane methods by [START_REF] Caffarelli | Asymptotic symmetry and local behavior os semilinear elliptic equations with critical Sobolev growth[END_REF], see also [START_REF] Li | Classification of solutions of some nonlinear elliptic equations[END_REF]. In the general case m > 1, the nonexistence of nontrivial solutions for p < p * 2 was proved in a beautiful article of Serrin and Zou [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF], then the extension to the case p = p * 2 was done by [START_REF] Sciunzi | Radial symmetry and applications for a problem involving the ∆ p (.) operator and critical nonlinearity in R N[END_REF] for m < 2, then [START_REF] Vétois | A priori estimates and application to the symmetry of solutions for critical p-Laplace equations[END_REF] for 1 < m < 2, and finally [START_REF] Sciunzi | Classification of the D 1,p (R N )-solutions of the critical p-Laplace equation in R N[END_REF] for any m > 1.

When p = 0, (1.1) reduces to the Hamilton-Jacobi equation

-∆ m u = |∇u| q .
The Liouville property was proved in [START_REF] Lions | Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre[END_REF] for m = 2, and in [START_REF] Bidaut-Véron | Keller-Osserman estimates for some quasilinear elliptic systems[END_REF] for any m > 1, using the Bernstein technique. In that case the nonexistence holds for any q > m -1, without any sign condition on the solution. Estimates of the gradient for more general problems can be found in [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF].

For the general case of equation (1.1), consider the range of exponents p > 0, p + q + 1 -m > 0.

As in the case q = 0, there exists a "first subcritical case", where

p < N (m -1) N -m - (N -1)q N -m ,
for which any supersolution in R N of equation (1.1) is constant, from [START_REF] Filippucci | Nonexistence of positive weak solutions of elliptic inequalities[END_REF]. Beyond this case, a second critical case appears when 0 ≤ q < m -1: indeed there exist radial positive nonconstant solutions of (1.4) whenever p ≥ p * m,q , where p * m,q = N (m -1) + m N -m -q((N -1)q -N (m -1) + m) (N -m)(m + 1 -q) , see [START_REF] Caristi | Nonexistence of solutions of quasilinear elliptic equations[END_REF] and [START_REF] Bidaut-Véron | Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient[END_REF]. When m = 2 < N and p > 0, equation

-∆u = u p |∇u| q (1.4)
was studied in [START_REF] Bidaut-Véron | Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient[END_REF] for 0 < q ≤ 2. The case q = 2 could be solved explicitely by a change of the unknown function, showing that the Liouville property holds for any p > 0. Using a direct Bernstein technique we obtained a first range of values of (p, q) for which the Liouville property holds, in particular it holds when p + q -1 < 4 N -1 , covering the first subcritical case. Using an integral Bernstein technique in the spirit of [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] we obtained a wider range of (p, q) ensuring the Liouville property, recovering Gidas and Spruck result p < N +2 N -2 when q = 0. However some deep questions remained unsolved: Does the property hold for any p < p * 2,q when q < 1 ? Does it hold for any p > 0 when 1 ≤ q < 2 ?

In a recent article, Filippucci, Pucci and Souplet [20, Theorem 1.1] considered the case m = 2, q > 2, of a superquadratic growth in the gradient, a case which was not covered by [START_REF] Bidaut-Véron | Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient[END_REF]. They proved the following:

Theorem [20, Theorem 1.1] Any classical positive and bounded solution of equation (1.4) in R N with q ≥ 2 and p > 0 is constant.

In this article, we prove that the Liouville property holds true not only for (1.4) but for the quasilinear equation (1.1) without the assumption of boundedness on the solution. Our main result is the following

Theorem 1.1 Let u be any positive C 1 (R N ) solution of equation (1.1), with 1 < m < N and q ≥ m, p ≥ 0. (1.5)
Then u is constant.

We show that the case q = m can still be solved explicitely, giving the complete behaviour of the solutions of the equation, see Theorem 2.1. Next we assume q > m. We prove that all the solutions in an exterior domain are bounded, and we give the asymptotic behaviour (|x| → 0 and |x| → ∞) of the solutions in R N \ {0}:

Theorem 1.2 Assume 1 < m < N, q > m, p ≥ 0. Then any positive C 1 solution u of (1.1) in R N \B r 0 is bounded. If u is a non-constant solution, then |∇u| does not vanish for |x| > r 0 . Moreover any positive solution u in R N \ {0} satisfies lim |x|→∞ u(x) = l ≥ 0. (1.6)
If l > 0, there exist constants C 1 , C 2 > 0 such that for |x| large enough,

C 1 |x| N -m m-1 ≤ |u(x) -l| ≤ C 2 |x| N -m m-1 . (1.7)
Concerning the solutions in B r 0 \ {0} and in particular in R N \ {0} we proved an estimate of the gradient, showing that the solution is continuous up to 0 but the gradient is singular at 0: Theorem 1.3 Assume 1 < m < N, q > m, p ≥ 0. Any positive solution u in B r 0 \ {0} is bounded near 0, it can be extended as a continuous function in B r 0 , such that u(0) > 0, and for any

x ∈ B r 0 2 \ {0} |∇u(x)| ≤ C |x| -1 q-m+1 , (1.8)
where C = C(N, p, q, m, u). Finally

|u(x) -u(0)| ≤ C |x| q-m q-m+1 , (1.9)
near 0, where

C = C(N, p, q, m, u(0)). Moreover, if u is defined in R N \ {0} , then u(x) ≤ u(0) in R N \ {0} .
Note that the exponent involved in (1.8) is independent of p, actualy the solution behaves like a solution of the Hamilton-Jacobi equation

-∆ m u = c |∇u| q , (1.10)
with c = u p (0). Finally we make an exhaustive study of the radial solutions for q > m, showing the sharpness of the nonradial results. We reduce the study to the one of an autonomous quadratic polynomial system of order 2, following the technique introduced in [START_REF] Bidaut-Véron | A new dynamical approach of Emden-Fowler equations and systems[END_REF]. Compared to other classical techniques, it provides a complete description of all the positive solutions, in particular the global ones, without questions of regularity. We prove the following: Theorem 1.4 Assume 1 < m < N, q > m, p ≥ 0 and u is any positive non constant radial solution r → u(r) of (1.1) in an interval (a, b) ⊆ (0, ∞). (i) If a = 0, then u is bounded, decreasing and singular:

lim r→0 u = u 0 > 0, lim r→0 r u q-m+1 = a m,q u p 0 , a m,q = (N -1)q -N (m -1) q + 1 -m . (1.11)
And for given u 0 > 0, there exist infinitely many such solutions;

(ii) If b = ∞, then u admits a limit a limit l ≥ 0 at infinity and

lim r→∞ r N -m m-1 |u(r) -l| = k > 0. (1.12)
Furthermore, for given l > 0, c = 0 there exists a unique local solution near ∞, such that

lim r→∞ r N -m m-1 (u(r) -l) = c. (1.13)
(iii) For any u 0 > 0, there exist infinitely many solutions in (0, ∞), decreasing, such that lim r→0 u = u 0 , but a unique one, satisfying

lim r→0 u = u 0 and lim r→∞ u = 0. (1.14)
There exist infinitely many solutions defined on an interval (0, ρ), such that lim r→ρ u = 0, and an infinity such that lim r→ρ u = -∞. Finally, there exist an infinity of solutions in (ρ, ∞) such that lim r→ρ u = 0, and an infinity of solutions such that lim r→ρ u = ∞.

Note that Theorems 1.2 and 1.3 lead to the following natural question: are all the solutions in R N \ {0} radially symmetric? This is still an open problem, even in the case p = 0 of the Hamilton-Jacobi equation.

To conclude this paper, we improve another result of [START_REF] Filippucci | A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient[END_REF], where it was noticed that [20, Theorem 1.1] was still valid for p < 0, q ≥ 2. We prove here a much more general result covering the case p = 0. Theorem 1.5 Assume 1 < m < N, p ≤ 0 and p + q + 1 -m > 0. Then there exists a constant C = C(N, p, q, m) > 0 such that for any positive C 1 solution u of (1.1) in a bounded domain Ω,

|∇u(x)| ≤ C dist(x, ∂Ω) -1 q+1-m , ∀x ∈ Ω. If Ω = R N , then u is constant.
Let us give a brief comment on the analogous equation with an absorption term:

-∆ m u + u p |∇u| q = 0.

(1.15)

In the case m = 2, 0 < q < 2, a complete classification of the solutions with isolated singularities was performed in [START_REF] Ching | Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term Anal[END_REF]. A main contribution was recently given by the same authors in [START_REF] Ching | Gradient estimates for nonlinear elliptic equations with a gradientdependent nonlinearity[END_REF] where they obtained optimal estimates of the gradient for any 1 < m ≤ N , p, q ≥ 0, p + q -m + 1 > 0, still by the Bernstein method.

Our paper is organized as follows. In Section 2 we first treat the case q = m. In Section 3 we give the main ideas of our proofs when q > m = 2, and we introduce some tools for the general case q > m > 1. Our main theorems are proved in Section 4, and Section 5 is devoted to the radial case. The extension to the case p ≤ 0 is given in Section 6.

The case q = m

If q = m we can express explicitely the solutions of (1.1). We prove the following:

Theorem 2.1 Let 1 < m < N, p ≥ 0, q = m. Then (i) any C 1 positive solution in R N is constant;
(ii) any nonconstant positive solution in R N \B r 0 has a limit l at ∞ and

lim |x|→∞ |x| m-N m-1 |u -l| = c > 0;
(iii) any positive solution in B r 0 \ {0} extends as a continuous function in B r 0 , or satisfies

lim x→0 u p+1 |ln |x|| = (N -m)(p + 1) m -1 ; (2.1) (iv) any positive solution in R N \ {0} is radial.
Proof. We use a change of variable already considered in [START_REF] Abdelhamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF]: the equation takes the form

-∆ m u = β(u) |∇u| m , with β(u) = u p . (2.2)
We set γ(τ ) = ´τ 0 β(θ)dθ = τ p+1 p+1 , and

U (x) = Ψ(u(x)) = ˆu(x) 0 e γ(θ) m-1 dθ := ˆu(x) 0 e θ p+1
(p+1)(m-1) dθ.

(2.3)

A function u is a solution of (1.1) if and only if the above function U satisfies

-∆ m U = 0,
and if u is nonnegative not identically 0, U is m-harmonic and positive. Conversely, u is derived from U by

u(x) = Ψ -1 (U (x)) = ˆU(x) 0 ds 1 + g(s)
where

g(s) = ˆs 0 β(Ψ(w))dw = ˆs 0 Ψ p (w)dw. (2.4) (i) If u is a solution in R N of (2.
2), it is constant. Indeed any nonnegative m-harmonic functions U defined in R N is constant, from the Harnack inequality, see [START_REF] Reshetniak | Mappings with bounded deformation as extremals of Dirichlet type integrals[END_REF], [START_REF] Serrin | Local behaviour of solutions of quasi-linear equations[END_REF] and [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF]Theorem II].

(ii) If u is defined in R N \B r 0 , then U is bounded, it admits a limit L at ∞ and there holds

|U (x) -L| ≤ C |x| p-N
p-1 near ∞, see [START_REF] Avila | Asymptotics at infinity of solutions for p-Laplace equations in exterior domains[END_REF] for more general results. Clearly the same properties hold for u (with another limit). (iii) If u is defined in B r 0 \ {0}, it follows from [START_REF] Serrin | Local behaviour of solutions of quasi-linear equations[END_REF] that, either U extends as a continuous m-harmonic function in B r 0 , or it behaves like k |x| p-N p-1 near 0, so (2.1) holds. (iv) If u is a solution in R N \ {0}, it is proved in [START_REF] Kichenassamy | Singular solutions of the p-Laplace equation[END_REF] that U is radial and endows the form

U (x) = k |x| m-N m-1 + λ with k, λ ≥ 0.
Then u is radial, and, using (2.4), it has the expression

u(x) = ˆλ 0 ds 1 + g(s) + ˆk|x| m-N m-1 0 ds 1 + g(s -λ) .
3 Main arguments of the proofs 3.1 Ideas in the case m = 2

Before detailling the proof of Theorem 1.1. for q > m, we give an overview of it in the simple case of equation (1.4), with m = 2, p > 0, q > 2. We set u = v b , with b ∈ (0, 1), and obtain

-∆v = (b -1) |∇v| m v + b q-1 v s |∇v| q ,
with s = 1 -q + b(p + q -1). Next we explicit the equation satisfied by z = |∇v| 2 . Taking in account the Böchner formula and Cauchy-Schwarz inequality in R N ,

- 1 2 ∆z + 1 N (∆v) 2 + < ∇(∆v), ∇v) ≤ - 1 2 ∆z + (Hessv) 2 + < ∇(∆v), ∇v >= 0,
we get an estimate of the form, with universal constants C i > 0,

-∆z + C 1 v 2s z q ≤ C 2 z 2 v 2 + C 3 1 v < ∇z, ∇v > +C 4 v s z q-2 2 < ∇z, ∇v >, then -∆z + C 5 v 2s z q ≤ C 6 z 2 v 2 + C 7 |∇z| 2 z . (3.1)
Using the Hölder inequality we deduce,

-∆z + C 8 v 2s z q ≤ C 9 v - 2(q+2s) q-2 + C 7 |∇z| 2 z .
The crucial step is an estimate of Osserman's type in a ball B ρ valid for functions satisfying the inequality

-∆z + α(x)z k ≤ β(x) + d |∇z| 2 z in B ρ ,
where k > 1. This is proved in Lemma 3.1 below, and it asserts that

z(x) ≤ C(N, k, d) 1 ρ 2 max Bρ 1 α 1 k-1 + max Bρ β α 1 k in B ρ 2 .
Then we take b = q-2 p+q-1 , in the same spirit as in [START_REF] Filippucci | A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient[END_REF], so that B α is constant and α -1 (x) = v 2 (x). We obtain an estimate max

B ρ 2 |∇v| ≤ C max Bρ v ρ 1 q-1 + 1 . But any solution in R N satisfies for any ρ ≥ 1 max Bρ v ≤ v(0) + Cρ max Bρ |∇v| ≤ Cρ(1 + max Bρ |∇v|), (3.2)
which yields max

B ρ 2 |∇v| ≤ C((max Bρ |∇v|) 1 q-1 + 1).
Using the bootstrap method developped in [START_REF] Bidaut-Véron | Singularities in Hamiltonian elliptic systems with absorption terms[END_REF] and [START_REF] Bidaut-Véron | Keller-Osserman estimates for some quasilinear elliptic systems[END_REF] based upon the fact that 1 q-1 < 1, we deduce that |∇v| ∈ L ∞ (R N ). Note that the boundness of |∇v| had been obtained in [START_REF] Filippucci | A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient[END_REF] but under the extra assumption u ∈ L ∞ (R N ), an assumption that we get rid of. Returning to u = v b , it means that

-∆u = u p |∇u| q ≤ C |∇u| 2 u ,
and the same happens for u -l , where l = inf R N u. It implies that w l = (u -l) σ is subharmonic for σ large enough. Then from [START_REF] Bidaut-Véron | Keller-Osserman estimates for some quasilinear elliptic systems[END_REF], see also Lemma 3.3 below, and since u is superharmonic,

sup B R w l ≤ C 1 |B 2R | ˆB2R w 1 σ σ = C 1 |B 2R | ˆB2R (u -l) σ ≤ C (inf B R u -l) σ . Since C is independent of R, it follows that sup R N w l = 0, thus u ≡ l.
Next we consider a solution in an exterior domain and we replace (3.2) by a more precise comparison estimate between v(x) and its infimum on a sphere of radius |x| , and use the fact that this infimum is bounded as r → ∞. Then we can show that u is still bounded, and obtain the behaviour near ∞ by a careful study of u and w l . Finally we study the behaviour in B r 0 \ {0} by the Bernstein technique, not relative to v but directly to u, that means we take b = 1, so that s = p. From (3.1) the function ξ = |∇u| 2 satisfies

-∆ξ + C 5 u 2p ξ q ≤ C 6 ξ 2 u 2 + C 7 |∇z| 2 z ,
and k = inf B r 0 2
\{0} u is positive by the strong maximum principle, thus

-∆ξ + C 8 ξ q ≤ C 9 ξ 2 + C 7 |∇z| 2 z ≤ C 8 2 ξ q + C 11 + C 7 |∇z| 2 z ,
from what we deduce the estimates of ξ.

Some tools

In the sequel we use the Bernstein method. In the case p = 0, it appeared that the square of the gradient is a subsolution of an elliptic equation with absorption, for which one can find estimates from above of Osserman's type. In the case of equation (1.1), the problem is more difficult, but such upper estimates were also a main step in study of [START_REF] Bidaut-Véron | Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient[END_REF] of equation (1.4) for q < 2. Here also they constitue a crucial step of our proofs below. The following Lemma gives an Osserman's type property of such equations, extending of [8, Lemma 2.2], see also used in [7, Proposition 2.1].

Lemma 3.1 Let Ω be a domain of R N , and z ∈ C(Ω) ∩ C 2 (G), where G = {x ∈ Ω : z(x) = 0} . Let w → Aw= -N i,j=1 a ij ∂ 2 w
∂x i ∂x j be a uniformly elliptic operator in the open set G :

θ |ξ| 2 ≤ N i,j=1 a ij ξ i ξ j ≤ Θ |ξ| 2 , θ > 0. (3.3)
Suppose that for any x ⊂ G,

A(z) + α(x)z k ≤ β(x) + d |∇z| 2 z ,
with k > 1, and d = d(N, p, q), and α, β are continuous in Ω and α is positive. Then there exists c = c(N, p, q, k) > 0 such that for any ball B(x 0 , ρ) ⊂ Ω there holds

z(x 0 ) ≤ c 1 ρ 2 max Bρ(x 0 ) 1 α 1 k-1 + max Bρ(x 0 ) β α 1 k . Proof. Let B(x 0 , ρ) ⊂ Ω. We can assume that z(x 0 ) = 0. Let r = |x -x 0 | . Let w be the function defined in B ρ (x 0 ) by w(x) = λ(ρ 2 -r 2 ) -2 k-1 + µ, where λ, µ > 0. Let G 1 be a connected component of {x ∈ B ρ (x 0 ); z(x) > w(x)} . Then G 1 ⊂ G and G 1 ⊂ B(x 0 , ρ) ⊂ G. We define Lw in B ρ (x 0 ) by L(w) = A(w) + α(x)w k -β(x) -d |∇w| 2 w .
Then

w x i = 4λ k -1 (ρ 2 -r 2 ) -2 k-1 -1 x i , w x i x j = 4λ k -1 (ρ 2 -r 2 ) -2 k-1 -1 δ ij + 4λ(k + 1) (k -1) 2 (ρ 2 -r 2 ) -2 k-1 -2 x i x j , A(w) = - N i,j=1 a ij w x i x j = 4λ k -1 (ρ 2 -r 2 ) -2 k-1 -1 (- N i,j=1 a ij δ ij ) + 4λ(k + 1) (k -1) 2 (ρ 2 -r 2 ) -2 k-1 -2 (- N i,j=1 a ij x i x j ) ≥ -Θ( 4λN k -1 (ρ 2 -r 2 ) -2 k-1 -1 + 4λ(k + 1) (k -1) 2 (ρ 2 -r 2 ) -2 k-1 -2 r 2 = -Θ( 4λN k -1 (ρ 2 -r 2 ) -2 k-1 -2 (N (ρ 2 -r 2 ) + k + 1 k -1 r 2 ) = -Θ( 4λ k -1 (ρ 2 -r 2 ) -2 k-1 -2 (N ρ 2 + ( k + 1 k -1 -N )r 2 ), |∇w| 2 = 16λ 2 (k -1) 2 (ρ 2 -r 2 ) -4 k-1 -2 r 2 =⇒ |∇w| 2 w ≤ 16λ (k -1) 2 (ρ 2 -r 2 ) -2 k-1 -2 r 2 ,
and

w k = (λ(ρ 2 -r 2 ) -2 k-1 + µ) k ≥ µ k + λ k (ρ 2 -r 2 ) -2k k-1 = µ k + λ k (ρ 2 -r 2 ) -2 k-1 -2 .
We deduce from this series of inequalities,

L(w) ≥ α(x)µ k -β(x) + λ(ρ 2 -r 2 ) -2k k-1 (λ k-1 C(x) -Θ( 4 k -1 (N ρ 2 + ( k + 1 k -1 -N )r 2 ) - 16dr 2 (k -1) 2 ≥ α(x)µ k -β(x) + λ(ρ 2 -r 2 ) -2k k-1 (λ k-1 C(x) -c ρ 2 ), where c = Θ( 4 k-1 (2N + k+1 k-1 ) + 16d (k-1) 2 = c (N, p, q, k).
We deduce that L(w) ≥ 0 if we impose

µ k ≥ max Bρ(x 0 ) β α and λ k-1 ≥ c ρ 2 max Bρ(x 0 ) 1 α . If x 1 ∈ G 1 is such that z(x 1 ) -w(x 1 ) = max G 1 (z -w) > 0, then ∇z(x 1 ) = ∇w(x 1 ), and A(z - w)(x 1 ) ≥ 0. Therefore 0 ≥ L(z -w)(x 1 )) = A(z -w)(x 1 ) + α(x)(z k -w k )(x 1 ) + d |∇w| 2 w - |∇z| 2 z .
Since the last term is positive, it is a contradiction. Then z ≤ w in B ρ (x 0 ). In particular z(x 0 ) ≤ w(x 0 ).

We also use a bootstrap argument, initialy used in [11, Lemma 2.2], and then in [START_REF] Bidaut-Véron | Keller-Osserman estimates for some quasilinear elliptic systems[END_REF] in more general form. Lemma 3.2 Let d, h ∈ R with d ∈ (0, 1) and y be a positive nondecreasing function on some interval (r 1 , ∞). Assume that there exist K > 0 and ε 0 > 0 such that, for any ε ∈ (0, ε 0 ] and r > r 1 , y(r) ≤ Kε -h y d (r(1 + ε)).

Then there exists

C = C(K, d, h, ε 0 ) such that sup (r 1 ,∞) y ≤ C. Proof. Consider the sequence {ε n } = {ε 0 2 -n } n≥1 . Since the series ε n is convergent, the sequence {P m } := { m i=1 (1 + ε i )} m≥1
is convergent too, with limit P > 0. Then there holds for any r > r 1 y(r) ≤ Kε -h 1 y d (r(1 + ε 1 )) = Kε -h 1 y d (rP 1 ). We deduce by induction,

y(r) ≤ K 1+d+..+d m (ε -h 1 ε -hd 2 ..ε -hd m-1 m )y d m (P m r) = (Kε -h 0 ) 1+d+..+d m (2 h(1+2d+..+md m-1 ) )y d m (P m r),
and rP m → rP, d m → 0, thus (y(P m r)) d m → 1. Therefore we deduce that for any r > r 1 ,

y(r) ≤ (Kε -h 0 ) ∞ m=1 d m 2 ∞ m=1 md m-1 = (Kε -h 0 ) 1 1-d 2 d (1-d) 2 .
We also mention below a property of m-subharmonic functions given in [9, Lemma 2.1]. It's proof is also based upon a boostrap method and is valid for more general quasilinear operators:

Lemma 3.3 Let u ∈ W 1,m
loc (Ω) be nonnegative, m-subharmonic function in a domain Ω of R N . Then for any τ > 0, there exists a constant C = C(N, m, τ ) > 0 such that for any ball B 2ρ (x 0 ) ⊂ Ω and any ε ∈ 0, 1 2 ,

sup Bρ(x 0 ) u ≤ Cε -N m 2 τ 2 1 B (1+ε)ρ (x 0 ) ˆB(1+ε)ρ (x 0 ) u τ 1 τ
.

Finally we use some simple properties of mean value on spheres of m-superharmonic functions, in the same spirit as the ones given in [2, Lemmas 3.7, 3.8, 3.9] for mean values on annulus, and in [START_REF] Burgos-Perez | Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems[END_REF] for m = 2. For the sake of completeness we recall their proofs.

Lemma 3.4 Let u ∈ C 1 (Ω) be nonnegative, m-superharmonic in Ω. (i) If Ω = R N \B r 0 , then r → µ(r) := inf |x|=r u,
is bounded in (r 0 , ∞), and strictly monotone or constant for large r.

(ii) If Ω = B r 0 \ {0} , then r → µ(r) is nonincreasing in (0, r 0 ). Proof. (i) Let r > r 0 be fixed. The function f (x) = µ(r)(1 -( |x| r 0 ) m-N m-1
) is m-harmonic, and u ≥ f

on ∂B r ∪ ∂B r 0 , therefore u ≥ f in B r \B r 0 . Let k > 0 large enough such that 1 -k p-N p-1 ≥ 1 2 .
If we take r > kr 0 and any x such that |x| = kr 0 we obtain

u(x) ≥ µ(kr 0 ) ≥ f (x) = µ(r)(1 -k p-N p-1 ) ≥ 1 2 µ(r),
so µ(r) is bounded for r > kr 0 . For any r 2 > r 1 > r 0 , ϕ(r 1 , r 2 ) := inf Br 2 \B 1 u = min(µ(r 1 ), µ(r 2 )) from the maximum principle. Then ϕ is nonincreasing in r 2 and nondecreasing in r 1 . If µ has a strict local minimum at some point r, then for 0 < δ < δ 0 small enough, µ(r) < ϕ(r -δ 0 , r In both case we obtain a contradiction. Hence µ is either strictly monotone for r large enough, or it is constant, and so is u.

+ δ 0 ) ≤ ϕ(r -δ, r + δ),
(ii) For given r 1 < r 0 , and δ > 0, there exists ε δ ≤ r 1 such that for 0 < ε < ε δ , such that δε

m-N m-1 ≥ µ(r 1 ). Let h(x) = µ(r 1 ) -δ |x| m-N m-1 . Then u ≥ h on ∂B r 1 ∪ ∂B ε , then u ≥ h in B r 1 \B ε . Making ε → 0 and then δ → 0, one gets u ≥ µ(r 1 ) in B r 1 \ {0} , thus µ(r) ≥ µ(r 1 ) for r < r 1 .
4 Proof of the main results Then the operator

w → A(w) = -∆w -(m -2) D 2 w(∇v, ∇v) |∇v| 2 = - N i,j=1 a ij v x i x j , (4.1)
with coefficients a i,j depending on ∇v, is uniformly elliptic in G, and for any ε > 0, there exists

C ε = C ε (N, m, p, q, b, ε) such that - 1 2 A(z) + 1 -ε N (b -1) 2 (m -1) 2 -(1 -b)(m -1) z 2 v 2 + 1 -2ε N |b| 2(q-m+1) v 2s z q+2-m + 1 N 2(b -1)(m -1) -s |b| q-m bv s-1 z q+4-m 2 ≤ C ε |∇z| 2 z . (4.2) Proof. The following identities hold if u = v b : ∇u = bv b-1 ∇v, |∇u| m-2 ∇u = |b| m-2 bv (b-1)(m-1) |∇v| m-2 ∇v, ∆ m u = |b| m-2 b(v (b-1)(m-1) ∆ m v + (b -1)(m -1)v (b(m-1)-m |∇v| m ), -v (b-1)(m-1) ∆ m v = (b -1)(m -1)v (b(m-1)-m |∇v| m + |b| q v bp+(b-1)q |∇v| q ,
and finally

-∆ m v = (b -1)(m -1) |∇v| m v + |b| q-m bv s |∇v| q , (4.3)
with s = m -1 -q + b(p + q -m + 1). (4.4)

We set z = |∇v| 2 . Then in G,

-∆ m v = f ⇐⇒ -∆v -(m -2) D 2 v(∇v, ∇v) |∇v| 2 = f |∇v| 2-m ,
from which identity we infer

-∆v = (m -2) D 2 v(∇v, ∇v) |∇v| 2 + (b -1)(m -1) |∇v| 2 v + |b| q-m bv s |∇v| q+2-m where < Hess v(∇v), ∇v >= D 2 v(∇v, ∇v) = 1 2 < ∇z, ∇v > .
We recall the Böchner formula combined with Cauchy-Schwarz inequality,

- 1 2 ∆z + 1 N (∆v) 2 + < ∇(∆v), ∇v) ≤ - 1 2 ∆z + (Hess v) 2 + < ∇(∆v), ∇v >= 0. Since -∆v = m -2 2 < ∇z, ∇v > z + (b -1)(m -1) z v + |b| q-m bv s z q+2-m 2 , we deduce < ∇(∆v), ∇v >= - m -2 2 < ∇ < ∇z, ∇v > z , ∇v > +(1 -b)(m -1) < ∇ z v , ∇v > -|b| q-m b(sv s-1 z q+4-m 2 + q + 2 -m 2 v s z q-m 2 < ∇z, ∇v >).
we observe that

< ∇ z v , ∇v >= < ∇z, ∇v > v - z 2 v 2 and < ∇z, ∇v > 2 z 2 ≤ |∇z| 2 z , thus - m -2 2 < ∇ < ∇z, ∇v > z , ∇v >= - m -2 2 D 2 z(∇v, ∇v) z + 1 2 |∇z| 2 z - < ∇z, ∇v > 2 z 2 ≥ - m -2 2 D 2 z(∇v, ∇v) z -|m -2| |∇z| 2 z .
We define the operator A by (4.1); it satisfies (3.3) with θ = min(1, m -1) and Θ = max(1, m -1), so it is uniformly elliptic in G. Therefore

- 1 2 A(z) + 1 N (∆v) 2 -(1 -b)(m -1) z 2 v 2 -|b| q-m bsv s-1 z q+4-m 2 ≤ (b -1)(m -1) < ∇z, ∇v > v + (q + 2 -m) |b| q-m b 2 v s z q-m 2 < ∇z, ∇v > + |m -2| |∇z| 2 z . (4.5)
For ε > 0 there holds by Hölder's inequality,

q + 2 -m 2 v s z q-m 2 < ∇z, ∇v >≤ ε N |b| 2(q-m+1) v 2s z q+2-m + C ε |∇z| 2 z , (∆v) 2 = m -2 2 < ∇z, ∇v > z + (b -1)(m -1) z v + |b| q-m bv s z q+2-m 2 2 ≥ (b -1) 2 (m -1) 2 z 2 v 2 + |b| 2(q-m+1) v 2s z q+2-m + 2(b -1)(m -1) |b| q-m bv s-1 z q+4-m 2 -(m -2) |∇z| √ z (|b -1| (m -1) z v + |b| q-m+1 v s z q+2-m 2
), and for any ε > 0,

(m -2) |∇z| √ z (|b -1| (m -1) z v ≤ ε N (b -1) 2 (m -1) 2 z 2 v 2 + C ε |∇z| 2 , z (m -2) |∇z| √ z |b| q-m+1 v s z q+2-m 2 ≤ ε N |b| 2(q-m+1) v 2s z q+2-m + C ε |∇z| 2 z ,
thus (4.2) follows.

Proof of Theorem 1.1. We use Lemma 4.1 with b ∈ (0, 1), combined with the estimate

1 N 2(b -1)(m -1) -s |b| q-m bv s-1 z q+4-m 2 ≤ ε N |b| 2(q-m+1) v 2s z q+2-m + C ε z 2 v 2 .
Then there exist constants C i > 0 depending only on m, b, N, p, q, such that

1 2 A(z) + C 1 v 2s z q+2-m ≤ C 2 z 2 v 2 + C 3 |∇z| 2 z .
Next we choose s = -1 in (4.4), thus

b(p + q -m + 1) = q -m,
which is positive because q > m. We deduce using Hölder inequality,

A(z) + C 4 z q+2-m -C 5 v 2 ≤ A(z) + C 1 z q+2-m -C 2 z 2 v 2 ≤ C 3 |∇z| 2 z .
If we apply Lemma 3.1 with

α(x) = C 4 v 2 (x) , β(x) = C 5 v 2 (x) , k = q + 2 -m,
we deduce that any solution in B ρ (x 0 ) ρ > 0, satisfies

z(x 0 ) ≤ C 6 1 αρ 2 1 k-1 + C 5 C 4 1 k ≤ C 7 max Bρ(x 0 ) v ρ 2 q+1-m + 1), which yields |∇v(x 0 )| ≤ C 8 max Bρ(x 0 ) v ρ 1 q+1-m + 1 , (4.6)
where we observe that 1 q+1-m < 1, since q > m. Let ε ∈ 0, 1 2 . As a consequence, for any solution in B 2R , (or even B 3R 2 ) considering any x 0 ∈ B R and taking ρ = Rε, we get max

B R |∇v| ≤ c max BR(1+ε) v εR 1 q+1-m + 1 ≤ cε -1 q+1-m max BR(1+ε) v R 1 q+1-m + 1 , (4.7)
max

BR(1+ε) v ≤ v(0) + R(1 + ε) max BR(1+ε) |∇v| , max BR(1+ε) v R ≤ 1 + v(0) R + (1 + ε) max BR(1+ε) |∇v| ≤ c 0 1 R + max BR(1+ε)
|∇v| ,

where c 0 = 2 + v(0) depends on v(0). If R ≥ 1, max BR(1+ε) v R 1 q+1-m + 1 ≤ c 1 q+1-m 0 1 + max BR(1+ε) |∇v|) 1 q+1-m + 1 ≤ c 1 1 + max BR(1+ε) |∇v| 1 q+1-m .
Then from (4.7),

y(R) := 1 + max B R |∇v| ≤ 1 + c 2 ε -1 q+1-m (1 + max BR(1+ε) |∇v|) 1 q+1-m ≤ c 3 ε -1 q+1-m (1 + max BR(1+ε) |∇v|) 1 q+1-m .
Using the definition of y, this is

y(R) ≤ cε -1 q+1-m (y((1 + ε)R)) 1 q+1-m ,
where c depends on v(0). Therefore y(R) is bounded as a consequence of Lemma 3.2. Thus |∇v| is bounded and using the definition of v with the value of b,

|∇v| q-m = u p+1 |∇u| q-m ∈ L ∞ (R N ).
Next we consider any l ≥ 0 such that u -l > 0. The function

u l = u -l satisfies 0 ≤ -∆ m u l ≤ C ∞ |∇u| m u ≤ C ∞ |∇u l | m u l , with C ∞ = u p+1 |∇u| q-m L ∞ (R N )
. Then the function w l = u σ l with σ > 1 to be specified below, satisfies

-∆ m w l = σ m-1 u l (σ-1)(m-1) (-∆ m u l + (σ -1)(m -1) |∇u l | m u l ) ≤ σ m-1 ((σ -1)(m -1) -C ∞ )u σ(m-1)-m l |∇u l | m .
Therefore w l is m-subharmonic for σ large enough. We first take l = 0, so w = u σ . By Lemma 3.3, for any τ > 0, there exists a constant

C τ = C τ (N, m, τ ) such that sup B R w ≤ C τ 1 |B 2R | ˆB2R w τ 1 τ = C τ 1 |B 2R | ˆB2R u τ σ 1 τ , (4.8)
and since u is m-superharmonic, there holds for any θ ∈ (0, N (m-1) N -m ), [START_REF] Trudinger | On harnack type inequality and their applications to quasilinear elliptic equations[END_REF] inf

B R u ≥ c θ 1 |B 2R | ˆB2R u θ 1 θ . (4.9) Taking τ = θ σ , we deduce sup B R u = (sup B R w) 1 σ ≤ C 1 σ τ 1 |B 2R | ˆB2R u τ σ 1 sσ ≤ C 1 σ τ c θ inf B R u. (4.10)
This means that u, and also w, satisfies the Harnack inequality in R N :

sup B R w ≤ C τ c σ θ inf B R w.
But r → µ(r) = inf |x|=r u = inf Br u from the maximum principle, is nonincreasing, so it has a limit L ≥ 0 as r → ∞. This implies that u is bounded and l = inf R N u ≥ 0. If we replace u by u l and w by w l , then (4.8) holds with w and u replaced respectively by w l and u l since w l is m-subharmonic, but also (4.9) holds with u replaced by u l since u l is m-superharmonic. Thus sup

B R w l ≤ C(inf B R u l ) σ .
Therefore sup B R w l tends to 0 as R → ∞. Then w l ≡ 0, thus u ≡ l.

Asymptotic behaviour near ∞

In this section we consider the behaviour of solutions defined in an exterior domain.

Proof of Theorem 1.2. Consider a nonnegative solution

u = v b (0 < b < 1) of (1.1) in R N \B r 0 . From (4.6) the function v satisfies in B ρ (x 0 ) (ρ > 0), |∇v(x 0 )| ≤ C max Bρ(x 0 ) v ρ 1 q+1-m + 1 , (4.11)
with C = C(N, p, q, m). Here we denote by c i some positive constants depending on r 0 , N, p, q, m.

Let R > 4r 0 and 0 < ε ≤ 1 4 . Applying (4.11) with ρ = εR, we get

|∇v(x 0 )| ≤ c 1 max B εR (x 0 ) v εR 1 q+1-m + 1 ≤ c 1 ε -1 q+1-m max B εR (x 0 ) v R 1 q+1-m + 1 , then max |x|=R |∇v| ≤ c 2 max R(1-ε 2 )≤|x|≤R(1+ ε 2 ) v εR 1 q+1-m + 1 ≤ c 3 max R 1+ε ≤|x|≤R(1+ε) v εR 1 q+1-m + 1 , max R 2 ≤|x|≤2R |∇v| ≤ c 4   max R 2(1+ε) ≤|x|≤2R(1+ε) v εR 1 q+1-m + 1   ,
and finally,

1 + max R 2 ≤|x|≤2R |∇v| ≤ c 5 ε -1 q+1-m   max R 2(1+ε) ≤|x|≤2R(1+ε) v R 1 q+1-m + 1   . From Lemma 3.4-(i), µ(r) = inf |x|=r u = (inf |x|=r v) b is bounded : let M = max r≥r 0 µ(r)
. Note that M depends on u. Now for any x such that |x| = ρ, there exists at least one point x ρ where v(x ρ ) = inf |x|=ρ v . We can join any point x ∈ S ρ to x ρ by a connected chain of balls of radius ερ with at points x i ∈ S ρ and this chain can be constructed so that it has at most π ε elements. Considering one ball containing x and joining it to a ball containing x ρ , we get that

v(x) ≤ v(x ρ ) + C N ε -1 ρ max ρ 1+ε ≤|x|≤ρ(1+ε) |∇v| ≤ M 1 b + C N ε -1 ρ max ρ 1+ε ≤|x|≤ρ(1+ε) |∇v| . Then max R 2(1+ε) ≤|x|≤2R(1+ε) v ≤ c 1 M 1 + ε -1 R max R 2(1+3ε) ≤|x|≤2R(1+3ε) |∇v| ≤ c 2 M ε -1 R 1 + max R 2(1+3ε) ≤|x|≤2R(1+3ε)
|∇v| ,

and 1 εR max R 2(1+ε) ≤|x|≤2R(1+ε) v ≤ c 3 M ε -2 1 + max R 2(1+3ε) ≤|x|≤2R(1+3ε)
|∇v| .

Using estimate (4.7) we obtain

1 + max R 2 ≤|x|≤2R |∇v| ≤ c 4 M ε -2 q+1-m 1 + max R 2(1+3ε) ≤|x|≤2R(1+3ε) |∇v| 1 q+1-m . (4.12)
Let {ε n } n≥1 be a positive decreasing sequence such that

P n := n j=1 (1 + ε j ) → 2 and Θ n := n j=1 ε d j j+1 → Θ > 0 when n → ∞. It is easy to find such sequences such that ε j ∼ 2 -j . For R 2 ≤ a < 2R ≤ b we set y(a, b) = 1 + max a≤|x|≤b |∇v| .
Then (4.12) with (a, b) = ( R 2 , 2R) and ε 1 = 3ε asserts that

y( R 2 , 2R) ≤ c 5 ε -h 1 y( R 2(1+ε 1 ) , 2R(1 + ε 1 )) d with h = 2 q + 1 -m and d = 1 q + 1 -m ∈ (0, 1). Applying (4.12) with (a, b) = ( R 2Pn , 2RP n ) we obtain y( R 2Pn , 2RP n ) ≤ c 5 ε -h n+1 y( R 2P n+1 , 2RP n+1 ) d .
By induction, we deduce

y( R 2 , 2R) ≤ c 1+d+d 2 +...+d n 5 Θ -h n y( R 2P n+1 , 2RP n+1 ) d n+1 . (4.13) Since y( R 2P n+1 , 2RP n+1 ) → y( R 4 , 4R), we obtain that 1 + max R 2 ≤|x|≤2R |∇v| ≤ c d 1-d 5 Θ -h := C(M, N, p, q, m). (4.14)
Then we conclude again that |∇v| is bounded for |x| ≥ 4r 0 , then in R N \B r 0 since we have assumed that u ∈ C 1 R N \B r 0 . We consider again the function w = u σ , for σ depending of r 0 , large enough so that (σ -1)(m -1) ≥ u p+1 |∇u| q-m L ∞ (R N \Br 0 ) . As in the proof of Theorem 1.1 we conclude that w is m-subharmonic in R N \ Br 0 . Hence u satisfies the Harnack inequality, using the estimate (4.10). Therefore, for any

R > 2r 0 , sup R 2 ≤|x|≤3 R 2 u ≤ C inf R 2 ≤|x|≤3 R 2 u.
Since u is is m-superharmonic, it follows by the strong maximum principle, that it cannot have any local minimum in R N \ Br 0 . Since u σ is m-subharmonic it cannot have any local maximum too, and u shares this property. As a consequence |∇u| does not vanish in R N \ Br 0 . The function r → µ is bounded by Lemma 3.4, hence u is also bounded by the above Harnack inequality. Finally µ(r) is monotone for large r, so it admits a limit l ≥ 0 when r → ∞.

If µ(r) is nonincreasing for r ≥ r 1 > r 0 , then u -l ≥ 0, so we can consider the function w l instead of w. Then max

R≤|x|≤2R w l ≤ C( inf R≤|x|≤2R (u -l)) σ ,
Then w l tends to 0, thus u tends to l as |x| → ∞. Since u -l is m-superharmonic in R N \B r 0 , then there holds

u(x) -l ≥ C |x| m-N m-1 ,
with C = C(r 0 , N, m, u), see for example [5, Proposition 2.6], [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF]Lemma 2.3]. It is the case in particular when u is a solution in R N \ {0}. Note that the radial solutions such that µ is nonincreasing are precisely defined in (0, ∞). Now, it follows from the upper estimate of y(R), that the function u satisfies

-∆ m u = u p |∇u| q ≤ C |∇u| m u , in R N \B r 0 . Next suppose that l > 0. Then -∆ m u ≤ C |∇u| m .
The function U (still used in case q = m), defined by

U = (m -1)(e u-l m-1 -1), satisfies -∆ m U ≤ 0 and U tend to 0 at ∞. Then there exists R ε > 0 such that U (x) ≤ ε for |x| ≥ R ε . For R > R ε , the function x → ω(x) := ε + (sup |z|=r 0 U (z))( |x| r 0 ) m-N m-1 is a m-harmonic in B R \B r 0 , hence it is larger than U . Letting ε → 0 we get U ≤ C |x| m-N
m-1 near ∞; and U has the same behaviour as u -l , so we deduce the estimate from above,

u(x) -l ≤ C |x| m-N m-1 .
Then we get the estimate (1.7). Remark 4.2 (i) In case u is defined in R N \ {0} and l = 0, we obtain the estimates

C 1 |x| m-N m-1 ≤ u(x) ≤ C 2 |x| 1 σ m-N m-1 .
It would be interesting to improve the estimate from above. (ii) If u is defined in R N \B r 0 and if µ is nonincreasing, we have proved that u has a limit l ≥ 0 as |x| → ∞. If µ is nondecreasing, we only obtain that µ(r) = inf |x|=r u(x) has a limit l, and sup |x|=r u(x) has a limit λ ≥ l. Indeed the function w is m-subharmonic positive and bounded, so the function r → sup |x|=r w = (sup |x|=r u) σ is also monotone for large r and has a limit λ σ . We have w = u σ ≤ λ σ , so sup |x|=r u is also nondecreasing. But we cannot prove that λ = l.

Behaviour near an isolated singularity

In this section we study the behaviour of solutions with an isolated singularity at the origin.

Proof of Theorem 1.3 Let u be a nonnegative solution u of (1.1) in B r 0 \ {0} . We apply directly the Bernstein method to u : we obtain by Lemma 4.1 with b = 1, and then s = p. Setting ξ = |∇u| 2 , we get

1 2 A(ξ) + C 1 u 2p z q+2-m ≤ C 2 ξ 2 u 2 + C 3 |∇ξ| 2 ξ .
By the strong maximum principle, there exists a constant a 0 > 0 depending on r 0 and N, p, q, such that u ≥ a 0 in B r 0 2 \ {0} . Therefore, there holds

1 2 A(ξ) + C 2p 1 a 2p 0 z q+2-m ≤ C 2 ξ 2 a 2 0 + C 3 |∇ξ| 2 ξ , in B r 0 2 \ {0}.
Then from Lemma 3.1, we deduce the inequality

z(x 0 ) ≤ c   1 a 2p 0 ρ 2 1 q+1-m + 1 a 2(p+1) 0 1 q+2-m   ≤ c 2 0 1 ρ 2 q+1-m + 1 , for any ball B ρ (x 0 ) ⊂ B r 0 2 \ {0}, with c = c(N, p, q, m) and c 2 0 = c a -p q+1-m 0 + a -p+1 q+2-m 0 . Hence for any x ∈ B R \ {0} , with R ≤ min(1, r 0 8 ), |∇u(x)| ≤ 2c 0 |x| 1 q+1-m ≤ 2c 0 |x| 1 q+1-m . As a consequence, considering any x R such that x, x ∈ B R 2 \ {0}, there holds u(x ) -u(x) ≤ 2c 0 R q-m q+1-m .
Since q > m, u is bounded near 0. Then, with constants C > 0 depending on a 0 ,

-∆ m u = f ≤ C |∇u| q ≤ C |x| - q q+1-m . Then f ∈ L N m +ε loc (B R 4 ), since N -N m q q+1-m = N (m-1)(q-m)
m(q+1-m) > 0. Thus from [START_REF] Serrin | Local behaviour of solutions of quasi-linear equations[END_REF] u can be extended as a continuous function, solution of the equation in the sense of distributions. Then we deduce that for any

x ∈ B R 2 \ {0}, |u(0) -u(x)| ≤ c 0 |x| q-m q+1-m .
Moreover, replacing r 0 by ρ > 0 small enough such that u(x) ≥ u(0) 2 in B ρ , then a 0 ≥ u(0) 2 , hence c 0 ≤ C(N, p, q, m, u(0)), and for |x| ≤ min(1, ρ 8 ), we infer

|u(0) -u(x)| ≤ C |x| q-m q+1-m .
Next assume that u is defined in R N \ {0} and is not constant. Then u is bounded, since it is bounded near 0 and ∞. Then r → µ(r) = inf |x|=r u is nonincreasing, thus µ(r) ≤ u(0): indeed ∀ε > 0, we have µ(|x|) ≤ u(x) ≤ u(0) + ε for any |x| ≤ r ε , then from the monotone decreasingness, µ(r) ≤ u(0) + ε for any r > 0. From Theorem 1.2 lim |x|→∞ u = l = lim r→∞ µ. and then necessarily l ≤ u(0). Suppose that there exists x = 0 such that u(x) > u(0); then u has a maximum in R N \ {0} , but |∇u| cannot vanish in R N \ {0} by Theorem 1.2, so we get a contradiction.

So N 0 is a source; the particular trajectory X = 0 associated to λ 2 is not admissible. There exists an infinity of trajectories starting from N 0 as t → -∞, associated to the eigenvalue λ 1 ; the solutions (X, Z) satisfy X > 0, lim t→-∞ e -q-m q+1-m t X = C 0 , where C 0 > 0 is arbitrary and lim t→-∞ Z = a m,q ; then from (5.6) and the definition of Z , there exist infinitely many decreasing singular solutions u of (5.1) defined near 0, satisfying (1.11).

(ii) The linearisation at O gives the system

X t = -N -m m-1 X Z t = (N -N -1 m-1 q)Z,
with admits the eigenvalues ξ 1 , ξ 2 . So O is a sink, two particular trajectories are the axis X = 0 and Z = 0 which not admissible. There is an infinity of trajectories converging to O as t → ∞, tangent to the axis Z = 0, associated to the eigenvalue ξ 1 , with either X, Z > 0, or X, Z < 0 . They satisfy

X ∼ t→∞ C 1 e -N -m m-1 t , Z =∼ t→∞ C 2 e (N -N -1 m-1 q)t , with C 1 , C 2 > 0. ( 5.7)
The corresponding solutions u of (5.1) are defined for large r, and either decreasing or increasing; from (5.6), we obtain lim

r→∞ u = (C m-1-q 1 C 2 ) 1 p+q-m+1 = l > 0 and lim r→∞ r N -1 m-1 u = -lC 1 , thus lim r→∞ r N -m m-1 (u -l) = -lC 1 .
Thus (1.12) and (1.13) follow. The uniqueness property follows from the uniqueness of a trajectory satisfying (5.7) for given C 1 , C 2 , see also Remark 5.3 below.

(iii) Linearisation at A 0 : setting X = N -m m-1 + X, we get

X t = N -m m-1 (X + Z m-1 ) Z t = -(N -m)p+(N -1)q-(m-1)N m-1 Z,
which admits the eigenvalues µ 1 < 0 < µ 2 and the eigenvectors, with

d m,q = m -1 N -m (N -m + |µ 1 | (m -1)
It is a saddle point. The trajectory X = 0 associated to µ 2 is not admissible. Then a unique trajectory T A 0 converging to A 0 as t → ∞. By the scaling (5.3), we deduce the uniqueness property for u.

Next we a complete description of the local and global solutions in the phase-plane leading to the conclusions of Theorem 1.4: Proof of Theorem 1.4 when p > 0. We consider the sets

L X = {(X, Z) ∈ Q : X t = 0} = (X, Z) ∈ Q : X - N -m m -1 + Z m -1 = 0 , L Z = {(X, Z) ∈ Q : Z t = 0} = (X, X) ∈ Q : N - N -1 m -1 q -pX + q + 1 -m m -1 Z .
The straight line L X has an extremity at A 0 , with slope -(m -1), and the slope of

T A 0 is -d m,q < -(m -1), so T A 0 is above L X near t = ∞.
The line L X has an extremity at N 0 , with slope p(m-1) q+1-m , is located above L X for X > 0. The trajectories issued from N 0 have the slope c m,q , and we check that it is greater than p(m-1) q+1-m because q > 1 -m; so they start above L Z . (i) The trajectory T A 0 stays in the region R = 0 < X < N -m m-1 ; X -N -m m-1 + Z m-1 > 0 which is negatively invariant. Then X t stays positive, thus X is increasing, hence bounded. Either T A 0 stays under L Z , then Z t < 0 and Z is bounded, thus T A 0 converges to N 0 , or it crosses the line L Z at time t 0 , and for t < t 0 there holds Z t > 0 so that Z stays bounded, and T A 0 still converges to N 0 ; in fact the second eventuality holds, because of the slope of the eigenvector at N 0 . So the trajectory, T A 0 joins N 0 to A 0 . By scaling, for any u 0 > 0 there exists a unique solution u defined in (0, ∞) satisfying (1.14).

(ii) All the trajectories with one point in the bounded invariant region R delimitated by the axis X = 0, Z = 0 and T A 0 , join N 0 to O, and the corresponding solutions u are positive on (0, ∞), decreasing, and satisfy (1.11). The trajectories with one point in the region R ⊂ Q above T A 0 converge to N 0 as t → -∞, and satisfy X t > 0, since T A 0 is above L X , and cannot be bounded, since there is no fixed point in this region. They can be of two types:

• Either they cross L Z , then after crossing Z is decreasing, necessarily to 0; then from (5.6), u is defined in a maximal interval (0, ρ) with u(ρ) = 0. Such solutions exists because by any point on L Z passes a trajectory.

• Or they stay above L Z , thus Z increases to ∞; in this case from (5.6) u is defined in a maximal interval (0, ρ) with lim r→ρ u = -∞; Let us show the existence of such solutions: For given c > 0, we define L c = {(X, Z) ∈ Q : X > 0, Z = cX + a m,q } .

We compute the field on this line, and show that it is entering the region above L c for c large enough: indeed we obtain,

Z t X t -c = Z( q+1-m m-1 (Z -a m,q ) -pX) X(X -N -m m-1 + cX+am,q m-1 ) -c = Z(c q+1-m m-1 -p) X -N -m m-1 + cX+am,q m-1
-c > (c(q + 1 -m) -p(m -1))(cX + a m,q ) (m -1 + c)X + a m,q -(N -m) -c, and (c(q + 1 -m) -p(m -1))(cX + a m,q ) -c((m -1 + c)X + a m,q -(N -m))

= cX(c(q -m) -(p + 1)(m -1) -(p + a m,q (m -1)) + c(q -m)a m,q + N -m) -p(m -1)a m,q is positive for large c, since q -m > 0. All the solutions with one point above L c stay in this region, so above L Z , which proves the existence.

(iii) All the trajectories with one point in {(X, Z) ∈ Q : X < 0} satisfy X t > 0 from (5.5). Then X increases necessarily up to 0, and then Z t > 0 for large t, thus (X, Z) converges to O, and u is defined for r large enough, increasing and lim r→∞ u = l > 0.

• Either they cross L Z , then before crossing Z is decreasing, necessarily to 0; then from (5.6), u is defined in a maximal interval (0, ρ) with u(ρ) = 0. Such solutions exist as above.

• Or they stay under L Z , thus X and Z decrease to -∞; in this case from (5.6) u is defined in a maximal interval (0, ρ) with lim r→ρ u = -∞. Let us show their existence: for given k > 0 we compute the field on the line L k = {(X, Z) ∈ Q : X < 0, Z = kX} . On this line X t > 0 and Z t -kX t = Z(N -N -1 m-1 q -pX + q+1-m m-1 Z -X + N -m m-1 -Z m-1 ) = Z(( q-m m-1 -p+1 k )Z -N -1 m-1 (q -m)) = Z 2 ( q-m m-1 -p+1 k ) + N -1 m-1 (q -m) |Z| , is positive for large k. The region below L k is therefore negatively invariant, then the existence is folllows. This conclude the proof.

Remark 5.3 The change of variable u(r) = ũ(s), s = r m-N m-1 , introduced in [START_REF] Guedda | Local and Global Properties of Solutions of Quasilinear Elliptic Equations[END_REF], and also used in [START_REF] Burgos-Perez | Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems[END_REF] in case m = 2 < N, leads to the equation

(|ũ s | m-2 s ũ) s + m -N m -1 q-m s N -1
N -m (q-m) ũp |ũ s | q = 0.

(5.8)

Hence if u is not constant ũs does not vanish, from Remark 5.1, and (5.8) is equivalent to

(m -1)ũ ss + m -N m -1 q-m s N -1
N -m (q-m) ũp |ũ s | q-m+2 = 0.

(5.9)

In particular we find again the existence and uniqueness of local solutions near ∞, satisfying (1.13) for given l ≥ 0 and c = 0 (c > 0 if l = 0); indeed the problem reduces to the equation (5.9) with the initial conditions ũ(0) = l and ũs (0) = c.

6 The case p < 0

Proof of Theorem 1.5. We still consider u = v b , with b > 0 : we recall that from (4.3) (4.4)

-∆ m v = (b -1)(m -1) |∇v| m v + b q-m+1 v s |∇v| q ,
with s = 1 -q + m + b(p + q -m + 1). Next we take b = q + 1 -m p + q -m + 1 , thus here b ≥ 1 and s = 0, so,

-∆ m v = (b -1)(m -1) |∇v| m v + b q-m+1 |∇v| q , ( 6.1)
where the two terms have the same sign. Then z = |∇v| 2 satisfies

A(v) = -∆v - m -2 2 < ∇z, ∇v > z = (b -1)(m -1) z v + b q-m+1 z q+2-m 2
.

Setting , we get from (4.5)

- 1 2 A(z) + 1 N (∆v) 2 + (b -1)(m -1) z 2 v 2 ≤ (b -1)(m -1)( < ∇z, ∇v > v + q + 2 -m 2 b q-m+1 z q-m 2 < ∇z, ∇v >,
where now the term in z 2 v 2 has a positive coefficient. Since b ≥ 1, we get an estimate of the form

A(z) + C 1 z q+2-m ≤ C 3 |∇z| 2 z .
Since q + 2 -m > 1, we deduce the estimate in any ball B ρ (x 0 ),

|∇v(x 0 )| ≤ C 1 ρ 1 q+1-m
, from Lemma 3.1, where C is a universal constant, which leads to the conclusions.

  which yields a contradiction as δ → 0. Then µ is monotone. If it is constant on two intervals (a, b) and (a , b ) with b < a and non-constant on (b, a ) it follows by Vazquez's maximum principle [35] that u is constant on B b \ B a and on B b \ B a but non constant on B a \ B b . It means, always by Vazquez's maximum principle, -either min{µ(r) : a < r < b } = µ(a) (if µ is nondecreasing) and the minimum of u in B b \ B a is achieved in any point in B b \ B a , hence u is constant in B b \ B a , -or min{µ(r) : a < r < b } = µ(a) (if µ is nonincreasing) and the minimum of u in B b \ B a is achieved in any point in B b \ B a , hence u is constant in B b \ B a .

4. 1 Lemma 4 . 1

 141 Proof of the Liouville property for q > m We first give a general Bernstein estimate for solutions of equation (1.1): Let u be any C 1 positive solution of ((1.1) in a domain Ω, with m > 1 and p, q arbitrary real numbers. Let G = {x ∈ Ω : |∇u(x)| = 0} . Let u = v b with b ∈ R\ {0} and z = |∇v| 2 .

Radial case

If u is a positive radial solution of (1.1), and if we denote for simplicity u(r) = u(x) with r = |x|, then u satisfies the following o.d.e.

( u m-2 u ) + N -1 r u m-2 u + u p u q = r 1-N (r N -1 u m-2 u ) + u p u q = 0.

(5.1)

We begin with a simple observation about the set of zeros of u . We have shown above that any solution of the exterior problem is either constant, or its gradient does not vanish. In the radial case, the proof is elementary:

Then any nonnegative radial solution of (5.1) on a segment [r 1 , r 2 ] ⊂ (0, ∞) is constant, or strictly monotone.

Proof. By the strong maximum principle [START_REF] Vàzquez | A Strong Maximum Principle for Some Quasilinear Elliptic Equations[END_REF], we can assume that u > 0 on (r 1 , r 2 ). The function

is nonincreasing. Suppose that u has two zeros ρ 1 and ρ 2 in (r 1 , r 2 ), then by integrating W and using the equation, we deduce that u ≡ 0 on [ρ 1 , ρ 2 ], hence u is constant therein, therefore we can assume that [ρ 1 , ρ 2 ] is the maximal subinterval of [r 1 , r 2 ] where u vanishes. If [r 1 , ρ 1 ] = [r 1 , r 2 ], for example r 1 < ρ 1 , then u > 0 on (r 1 , ρ 1 ) where u (r) = (r 1-N W (r))

on (r 1 , ρ 1 ) and lim

Hence either u is constant or it is strictly monotone.

Next we make a complete description of the radial solutions for p ≥ 0, q > m.

The case p = 0

This case p = 0 of the Hamilton-Jacobi equation is well known, since equation (5.2) can be directly integrated, so the solutions are explicit, and are a Ariadne's thread for studying the case p > 0. We find different types of nonconstant solutions according to the sign of u :

where a m,q = (N -1)q-N (m-1)

m-1

> 0 since q > N (m-1) N -1 > m -1; and the value of u follows by integration, with the requirement that u > 0. The solutions such that C 1 > 0 satisfy lim r→0 r

, then lim r→0 u(r) = u 0 > 0 , since q > m. The conclusions of theorem 1.4 follow in that case.

The case p > 0

Equation (5.1) can be reduced to an autonomous system, since it is invariant by the transformation u → T λ u (λ > 0) given by T λ u(x) = λ -q-m p+q+1-m u(λx).

(5.3)

Here we perform a change of unknown, introduced in [START_REF] Bidaut-Véron | A new dynamical approach of Emden-Fowler equations and systems[END_REF], which consists in a differentiation of the equation, as in the Bernstein technique. We set

and obtain the following quadratic system of Kolmogorov type, valid any point t where u (t) = 0, and any reals m, p, q,

in the region Q = (X, Z) ∈ R 2 : XZ > 0 . Note that the trajectories X = 0 and Z = 0 are not admissible in our study. Since p + q = m -1, we can recover u and u by

The fixed points of the system in Q are

We begin by a local study of the different points and the correponding results for the solutions of (5.1): Lemma 5.2 (i) The point N 0 is a source, with eigenvalues 0 < λ 1 = q-m q+1-m < λ 2 = (N -1)q-N (m-1)

m-1

and eigenvectors v 1 = (1, c m,q ) with c m,q > 0 and v 2 = (0, 1). Then there exist infinitely many singular decreasing solutions u of (5.1) defined near 0, satisfying (1.11).

(ii) The point O is a sink, with eigenvalues 0

m-1 q, and eigenvectors u 1 = (1, 0) and u 2 = (0, 1). Then there exist infinitely many solutions u of (1.1) defined for large r and either increasing or decreasing near ∞, satisfying (1.12) and (1.13)

m-1 and eigenvectors w 1 = (1, -d m,q ) with d m,q > 0 and w 2 = (1, 0). Then for any c > 0 there exists a unique solution u of (5.1), defined at least for large r, such that lim r→∞ r N -m m-1 u = c > 0.

Proof. (i) We perform the linearization at N 0 : setting Z = a m,q + Z, we get, with X > 0 and Z > 0, X t = q-m q+1-m X Z t = a m,q (-pX + q+1-m m-1 Z), which gives the eigenvalues λ 1 , λ 2 and their respective eigenvectors, with the value of c m,q c m,q = p(N -1)q -N )(m -1) (N -1)q 2 -2N (m -1)q + (m -1)(N (m -1) + m) .


Edmunds Čižo
email: 1edmunds@chemi-pharm.com

Olga Lavrinenko
email: olga.lavrinenko@du.lv

Svetlana Ignatjeva
email: 3svetlana.ignatjeva@du.lv

OF THE EU COUNTRIES IN THE PERIOD 1995-2017 *

Keywords: financial development, EU, determinants of financial development

The aim of the research is to determine the impact of openness and political stability which characterize the state of political rights and civil liberties; financial state regulation; the determinant of legal traditions which determines judicial independence, impartiality of the courts, protection of property rights, etc., the determinant of financial institutions, as well as the impact of certain macroeconomic indicators on the financial development of the EU countries in the period 1995 -2017.

Introduction

There is a large number of factors which influence financial development; and specialized literature considers various approaches to the systematization of these factors [START_REF] Voghouei | A survey of the determinants of financial development[END_REF].

Determinant of openness. It has been determined that a significant growth in international trade and capital flows have a positive effect on financial development [START_REF] Zingales | The Weak Links[END_REF]. Huang and Temple [START_REF] Huang | Will political liberalisation bring about …financial development[END_REF][START_REF] Huang | Does external trade promote financial development?[END_REF] found that the increase in the level of market openness leads to the increase in financial depth. Moreover, countries with more advanced financial systems are more likely to increase exports of manufactured goods in GDP, and to increase exports of goods. Easing restrictions on international flows affects the liquidity of stock markets, which allows a larger number of foreign banks to operate [START_REF] Levine | Finance and growth: theory and evidence[END_REF], and has a positive impact on the performance of the domestic banking system.

Determinant of financial liberalization. Financial liberalization can be considered as the sum of the following components: privatization of public financial institutions and banks, a guarantee for a free entry into the financial sector and independence of the Central Bank, abolition of tools for monetary control and introduction of free tools for interest rate control [START_REF] Arestis | Financial liberalization and the relationship between finance and growth[END_REF][START_REF] Arestis | Financial development and economic growth: assessing the evidence[END_REF][START_REF] Arestis | The impact of financial liberalization policies on financial development evidence from developing economies[END_REF][START_REF] Caplinska | Analysis of financial literacy tendencies with young people[END_REF]. Liberalization of financial markets leads to better allocation of resources, and higher levels and efficiency of investment.

Other authors believe that financial development is affected by the prevailing form of ownership of credit institutions, as well as the features of the prevailing model of corporate ownership and management of financial institutions [START_REF] Novickytė | Banking consolidation as value creation to the buyer and the financial system (case of Lithuania)[END_REF]. It is more convenient to consider these factors from the viewpoint of the analysis of their configuration that determines the restriction of financial development and growth of economy (Čižo et al. 2018). Low banking capitalization and a low level of corporate management in banks were important features in the countries that had experienced financial liberalization. "Undercapitalized" banks have incentives to take on excessive risks, especially if additional "guarantees" of security (for example, a system of deposit insurance) have been introduced into the banking system. This phenomenon can be interpreted as a "defect" of the financial market. Minimization of its impact is associated with the establishment of an effective system of financial regulation and supervision. It should include strict requirements to maintenance of the amount of equity, increase in the transparency of banks' risk management systems, and disclosure of information about the relative exposure to large risks. It is possible to achieve the establishment of a better mechanism for financial intermediation via improving the corporate model, which implies a transition to the market control over banks' actions, and to the control by the stock market. In this case, it is really possible to connect the establishment of an effective system of bank's risk management with a system that allows "fining" bank shareholders who take excessive risk. This would entail the enhancement of market discipline of bank managers and could create conditions for sustainable growth.

Determinant of legal traditions. L´opez de Silanes (L´opez de Silanes et al. 1998) emphasizes the significance of a legal system for ensuring the protection of property rights. The relationship between the system of law and the degree of development of capital markets has been established in the works by La Porta, L´opez de Silanes et al., etc. Countries of the French branch of civil law (continental branch) have less developed capital markets, compared to the countries belonging to the Common Law system (the English branch). However, there are not any significant differences between groups of countries in respect of the development of the banking sector. Having studied legal rules and practice of rights protection of minority shareholders and creditors, the authors determined that the countries of the Common Law system create stronger protection for shareholders. Within the group of the code law, countries of the French civil law provide the worst legal protection for shareholders and creditors. The legal origin has a significant impact on the law enforcement practice. Countries of the Common Law system and Scandinavian countries of civil law have a better quality of the law enforcement practice, while 507 countries of the French civil law have the worst one. An interim conclusion drawn from these studies is that countries with a continental model of law, which presumably provide less legal protection to minority shareholders and creditors, have less developed capital markets and a greater concentration of ownership at the level of industries and firms. However, the importance of belonging to a particular legal tradition turned out to be less pronounced for the development of the banking system.

In addition, as Zingales [START_REF] Zingales | The Weak Links[END_REF]) mentioned, countries with the legal system based on the French civil code in 1913 and 1929 were at the same level of development as the Common Law countries. Their lagging behind started only after World War II. This could be a possible indication against the decisive influence of the factor of belonging to a certain legal tradition. Moreover, as Zingales showed, it is closely related to a set of historical, cultural, socio-economic, and political-economic factors. Therefore, it is extremely difficult to determine the priority and distinguish any of them.

Determinant which characterizes institutions.

There is a large number of interpretations of the concept "institution". North, D.C [START_REF] North | The Rise of the Western World --A New Economic History[END_REF][START_REF] North | Institutions, Institutional Change and Economic Performance[END_REF]) provided the following definition of institutions: "Institutions are the rules of the game of a society or more formally are the humanly devised constraints that structure human interaction". Hodgson [START_REF] Hodgson | What Are Institutions?[END_REF] believes that "institutions are the systems of established and prevalent social rules that structure social interaction". He states that organizations are specialized institutions. Dixit and Greif [START_REF] Dixit | Lawlessness and Economics: Alternative Modes of Governance[END_REF][START_REF] Greif | The Fundamental Problem of Exchange: A Research Agenda in Historical Institutional Analysis[END_REF] define institutions as "a system of social factorsrules, beliefs, norms, and organizations that guide, determine, and constrain human actions". This definition also included the concepts of an organization and institutions, presenting the organization as an example of institutions.

Institutions can be classified into two groups: 1. Formal or "strict" (public) institutions -universal and transferable (to other people) rules that include the Constitution, laws, statutes, regulations, and norms, as well as their elements such as the rule of law and property rights, strict compliance with the terms of the contract and the independence of regulatory systems (Amin 1999). 2. Informal or "soft" institutionsthey form the social capital of society and include a number of features of life in a certain society, such as norms, traditions, social conventions, interpersonal contacts, relationships, and informal groups, which are an important element for the formation of trust within society. Acemoglu, Johnson и Robinson (Acemoglu et al. 2005;[START_REF] Acemoglu | The Colonial Origins of Comparative Development: An Empirical Investigation[END_REF][START_REF] Acemoglu | Reversal of Fortune: Geography and Institutions in the Making of the Modern World Income Distribution[END_REF][START_REF] Acemoglu | Institutional Causes, Macroeconomic Symptoms: Volatility, Crises and Growth[END_REF]Acemoglu et al. 2005;[START_REF] Acemoglu | De Facto Political Power and Institutional Persistence[END_REF] studied formal institutions in more detail and presented the following interrelated classification: 1. Economic institutions. This group includes factors that determine the structure of incentives in society (i.e. incentives for economic agents to invest, accumulate production factors, make transactions, etc.) and ensure the redistribution of resources in the economy. For example, the structure of property rights, barriers to business, types of contracts formalized in legal documents, and tax transfer redistribution schemes. 2. Political power and political institutions. Economic institutions are the result of the collective choice of society. Society is made up of different groups with often opposite interests. The relative political power of these groups determines their ability to manage resources and implement economic policies. The distribution of political power determines the structure and quality of economic institutions. This, in turn, is a consequence of political power, i.e. the power which arises from economic results. Political institutions include institutions which delegate political power between various groups. They are related to the characteristics of the government and current constitution.

Voghouei [START_REF] Voghouei | A survey of the determinants of financial development[END_REF]) noted that political choice determines the forces of influence on the development and functioning of the financial system. Beck [START_REF] Beck | Law, Politics, and Finance[END_REF]) highlights that a financial system in the country under centralized, authoritarian, and closed political regime is likely to be less developed than in the country with democracy and an open, highly competitive government that is controlled by law. Political and

INSIGHTS INTO REGIONAL DEVELOPMENT

ISSN 2669-0195 (online) http://jssidoi.org/jesi/ 2020 Volume 2 Number 2 (June) http://doi.org/10.9770/IRD.2020.2.2(1) 508 economic factors directly affect financial development and have an indirect influence on other determinants of financial development. For example, an economic institution is one of the key factors of financial development. This factor is endogenous and it is determined by political institutions. Acemoglu (Acemoglu et al. 2005) claimed that different economic institutions lead to different distribution of resources. Siegle, Weinstein and Halperin (Siegle et al. 2004) suggest that democracy can be seen as a source for development. The authors believe that democratic systems are "always stronger" than authoritarian regimes; democracy and political freedom can promote financial development, as they support the establishment of such important institutions as mechanisms for checks and balances, a system of self-monitoring and restrictions based on regulations.

Voghouei [START_REF] Voghouei | A survey of the determinants of financial development[END_REF]) identified that political factors both directly affect financial development and play an important role in the formation of institutional openness.

Governments can pursue national economic goals, as well as narrow political interests by controlling banks' decisions regarding credit support for enterprises. This factor and a number of other factors prove that the privatization of banks can improve the efficiency of credit allocation and, therefore, have a positive effect on the quality and volume of investments. La Porta, Lopez-de Silanes, and Shleifer in their work provided the assessment of the impact of bank privatization on the economic growth (La [START_REF] Porta | Government Ownership of Banks[END_REF]. They discovered that government ownership of banks is negatively correlated with both financial development and growth. At the same time, a 10% decrease in the share of bank assets owned by the government is related to 0.25% increase in growth per year. In addition, they show that it is impossible to achieve a positive effect of growth if the privatization of banks happens without elimination of institutional defects (a weak protection of property rights and low government efficiency).

Governments do not always demonstrate their willingness to use the potential resource of financial development. A political and economic factor can explain a phenomenon of uneven financial development, i.e. the existence and power of a pressure group which either supports or opposes the idea of financial openness and financial development. The history of the 20 th century demonstrate that financial development can be neither provided nor be a success if the ruling elite opposes it [START_REF] Načisčionis | Secure development of public administration[END_REF]. Poor development of financial markets creates an environment for officials from large and well-known industrial or financial companies to benefit from monopolistic nature of rents. Companies' market power under a poor disclosure system and weak contracts imposes high entry barriers for new potential agents. Therefore, the management of such companies (having a strong influence in the government) often opposes financial openness and financial development.

Development level of the standards of financial reporting. The impact of accounting standards [START_REF] Subačienė | Evaluation of accounting regulation evolution in selected countries[END_REF]; Nadhir and Wardhani 2019), a bankruptcy system, and management standards and procedures on the growth and development of the financial sector was investigated by Levine and Beck, and others (Beck 2002;[START_REF] Beck | Law, Politics, and Finance[END_REF][START_REF] Levine | Finance and growth: theory and evidence[END_REF][START_REF] Puryaev | Mathematical tooling of accounting non-economic characteristics during the assessing process of investment project effectiveness[END_REF].

The authors found that countries with better accounting information, with an appropriate level of protection of investors' and creditors' rights, tend to have more intense development of financial intermediation. This way, the growth prospects are enhanced by the fact that the legal environment, formal and informal rules and procedures that structure the information environment and regulate the behaviour of corporate participants and investors, acting together, encourage the development of financial intermediation. Mechanisms for formation of changes and transfer of impulses to the economy provided by the financial sector, as well as the nature of the factors that determine the development of the financial sector are still relevant topics for research. In addition to Acemoglu's classification, Rodrik [START_REF] Acemoglu | The Colonial Origins of Comparative Development: An Empirical Investigation[END_REF][START_REF] Acemoglu | Reversal of Fortune: Geography and Institutions in the Making of the Modern World Income Distribution[END_REF][START_REF] Acemoglu | Institutional Causes, Macroeconomic Symptoms: Volatility, Crises and Growth[END_REF]Acemoglu et al. 2005;[START_REF] Acemoglu | De Facto Political Power and Institutional Persistence[END_REF][START_REF] Rodrik | Growth Strategies[END_REF] 1. Institutions of market formationinstitutions of judicial, legislative, and administrative state system which ensure property rights and execution of contracts. Guarantees for property rights and execution of contracts affect incentives for economic activity, contribute to increased productivity and resource efficiency. The incentive structure of economy becomes significantly limited without these institutions. 2. Institutions of market stabilizationinstitutions of monetary circulation and tax regulation, prudential regulation and supervision, institutions of risk coordination and sharing (for example, a banking and credit system, tax authorities, stock market, pension funds, private and state venture companies, and agencies). Countries need monetary, fiscal and other measures to combat economic cycles [START_REF] Katan | Ensuring complex security of the financial flows movement in the national economy system[END_REF]. Ensuring complex security of the financial flows movement in the national economy system, Journal of Security and Sustainability Issues 9(1): 39-50.. Therefore, the effectiveness of market stabilization institutions provides for consolidation of financial resources, helps to coordinate the work of small and medium-sized investors, allows more efficient allocation of resources in the economy, enables to reduce transaction costs, and to support the innovation process and successful emergence of new sectors in the economy. 3. Institutions of market regulationregulatory government bodies that provide control and monitoring of various aspects of enterprises' daily activities (opening, expansion, and liquidation of business; access to infrastructure, and land; issue of building permits; compliance with the general rules for foreign trade; licensing and certification; tax rules; sanitary and epidemiological regulations; safety regulations), and have the right to suspend enterprises' activity; institutions of antimonopoly regulation and control which ensure the quality of market competition. Market requires comprehensive rules to reduce possibilities for abusing market power, the influence of external factors, and combating information asymmetry, as well as creation of a product and security standards, etc. In general, the profitability of operations and attractiveness of private investment depend on the efficiency and impartiality of functions of market regulation institutions. 4. Institutions of market legitimizationinstitutions of human capital development (institutions in the sectors of healthcare, education, social security and insurance).

Institutions provide for appropriate environment for creativity, innovations, and protection of intellectual rights. In addition, they increase competition for potential opportunities (resources), at least as long as the rule of law applies to all members of society without exception. Developed institutions provide for protection of property rights; they also promote full consistency with contracts, integration into the global economy, and support for macroeconomic stability; they help manage financial intermediaries' risks, and provide for social insurance and social protection, as well as hold government accountable. These factors create incentives for economic agents to conclude and implement a larger number of long-term contracts, which eventually contributes to the increase in investment and sustainable economic growth. The quality of institutions has a positive impact on such components of the country's economic development as functions of its financial system, development of entrepreneurship, and inflow of foreign direct investment (FDI).

Macroeconomic determinants. Inflation, investments, and economic growth influence financial development. Huybens and Smith [START_REF] Huybens | Inflation, financial markets and long-run real activity[END_REF] determined that inflation has a negative impact on financial development. The higher the level of inflation is, the lower the real return on money is, which results in decrease in lending. If the financial sector provides less credit, it means that the distribution of money is inefficient, and has a negative effect on financial development. Levine [START_REF] Levine | Finance and growth: theory and evidence[END_REF] found that the gross national income per capita and the savings rate are positively linked to financial development. [START_REF] Stulz | Culture, openness, and finance[END_REF] identified the influence of cultural differences such as differences in language and religion on the level of financial development. The study confirmed that culture might be the factor that explains differences between countries regarding protection of investors' rights and provision of compliance with contracts, in particular depending on the rights of creditors. FID -financial institutions depth; FIA -financial institutions access; FIE -financial institutions efficiency; FMD -financial markets depth; FMA -financial markets access; FME -financial markets efficiency. FID -financial institutions depth is characterized by the following indicators: private-sector credit (% of GDP), pension fund assets (% of GDP), mutual fund assets (% of GDP), insurance premiums, life and non-life (% of GDP). FIA -financial institutions access is characterized by the following indicators: branches (commercial banks) per 100,000 adults; ATMs per 100,000 adults. FIE -financial institutions efficiency is characterized by the following indicators: net interest margin; lending-deposits spread; non-interest income to total income; overhead costs to total assets; return on assets; return on equity. FMD -financial markets depth is characterized by the following indicators: stock market capitalization to GDP; stocks traded to GDP; international debt securities government (% of GDP); total debt securities of nonfinancial corporations (% of GDP); total debt securities of financial corporations (% of GDP). FMA -financial markets access is characterized by the following indicators: percent of market capitalization outside of top 10 largest companies; total number of issuers of debt (domestic and external, nonfinancial corporations, and financial corporations). FME -financial markets efficiency is characterized by the following indicator: stock market turnover ratio (stocks traded/capitalization).

Culture and geography. Stulz and Williamson

Each indicator is standardized from 0 to 1. The lowest value of the indicator for countries is zero, and all other values are measured regarding this minimum value. In order to avoid the pitfalls appearing as a result of extreme data, the values of variables of the 5th and 95th percentile are defined as cut-off levels. Indicators are defined in a such way that higher values indicate better financial development. Then, the indicators are grouped into six subindices in the lower part of the pyramid (see Fig. above). The aggregation is a weighted average of the base series, where the weights are the squares of the factor loadings from the analysis of principal components, in such a way that their sum comprises 1. Finally, sub-indices are similarly aggregated into higher indices using the factor analysis according to the method of principal components; the FD index is aggregated in a similar way. Source: developed by the authors on the basis of the following sources: [START_REF] Voghouei | A survey of the determinants of financial development[END_REF][START_REF] Porta | Government Ownership of Banks[END_REF][START_REF] Acemoglu | The Colonial Origins of Comparative Development: An Empirical Investigation[END_REF][START_REF] Acemoglu | Reversal of Fortune: Geography and Institutions in the Making of the Modern World Income Distribution[END_REF][START_REF] Acemoglu | Institutional Causes, Macroeconomic Symptoms: Volatility, Crises and Growth[END_REF], Acemoglu et al. 2005[START_REF] Acemoglu | De Facto Political Power and Institutional Persistence[END_REF][START_REF] Rodrik | Growth Strategies[END_REF][START_REF] Huybens | Inflation, financial markets and long-run real activity[END_REF][START_REF] Levine | Finance and growth: theory and evidence[END_REF][START_REF] Stulz | Culture, openness, and finance[END_REF].

The table 1 below presents indicators which characterize determinants of financial development

In order to identify the impact of the abovementioned determinants on the financial development of the EU countries in the period 1995-2017, the authors use a correlation analysis; the strength of relationship is characterized by the value of the Pearson correlation coefficient.

Research results

In order to calculate the impact of financial liberalization on financial development, the Economic Freedom of the World index (Economic Freedom in the World 2019) published by Fraser Institute in Canada (Fraser Institute) was applied. The integral index consists of sub-indices: Size of Government (a level of state regulation) (EF1), Legal System and Property Rights (EF2), Sound Money (EF3), Freedom to Trade Internationally (EF4), and Regulation (EF5). These 5 areas are divided into 24 components, which in turn also consist of several indicators. In general, 42 variables are used to calculate the index. Their average value determines the level of each component. The assessment of the area is calculated as the average value of all components.

Having calculated correlation coefficients of the Economic Freedom index and the Financial Development index, it is determined that there is a linear dependence; from 1995 until 2000 it was very strong and positive; later, until 2005 the linear dependence was average and positive; after 2005, values of the correlation coefficients decreased gradually, and since 2012, the linear dependence became very weak and positive; in 2017 it became close to 0 (Figure 2). In the period 1995 -2017 the impact of the Economic Freedom index which characterizes financial liberalization gradually decreased from being strong to being very weak.

Size of Government index consists of 4 components: government consumption, transfers and subsidies, government enterprises and investment, top marginal tax rate, and top marginal income tax rate. The index shows to what extent countries rely on a political process when distributing resources, goods, and services. When government spending increases as compared to that of individuals, households, and enterprises, political decisionmaking replaces personal choice. Government consumption as a share of total consumption, and transfers and subsidies as a share of the GDP are the indicators of the size of government. When government consumption comprises a larger share of the total volume, political choice is replaced with personal choice. Similarly, when governments tax some people in order to provide transfers to others, they restrict people's freedom to keep what they earn. The third component measures the extent to which countries use private investment and enterprises rather than public investment. Governments and state-owned enterprises play by rules that are different from those of private enterprises: they do not depend on consumers and private investors, and often operate in protected markets. The fourth component reflects top marginal income tax rate and top marginal income and payroll tax rate, and the threshold at which these rates begin to apply. These two subcomponents are averaged to calculate the top marginal tax rate. High marginal tax rates, which apply at relatively low levels of income, also indicate the dependence on government. These rates deprive people of results of their work. Therefore, countries with low government spending as a share of the total volume, a smaller sector of state-owned enterprises and lower marginal tax rates have the highest ratings in this area.

There is a negative linear dependence between the values of the Financial Development index and the Government Regulation index: business regulations, labour market regulations, and credit market regulations (a subsystem of the Economic Freedom index), i.e. the larger the size of government is and the more it interferes in business, labour and credit markets, the poorer the financial development is. In 1995 the Pearson correlation coefficient comprised -0.052, after that it was gradually increasing, and by 2017 it comprised -0.482. Therefore, the influence of this factor on financial development during the period under study increases. At the same time, there is a danger that by controlling the decisions of banks regarding credit support for enterprises, government may pursue not only national economic goals, but also narrow political interests. Determinants of culture and geography will be not considered within the framework of the research as they do not belong to the sphere of economy.

Conclusions and discussion

In general, the correlation analysis of the impact of liberalization on financial development of the EU countries in the period 1995 -2017 confirms the ideas by Arestis, Demetriades, Bassam and Kostas [START_REF] Arestis | Financial liberalization and the relationship between finance and growth[END_REF][START_REF] Arestis | Financial development and economic growth: assessing the evidence[END_REF][START_REF] Arestis | The impact of financial liberalization policies on financial development evidence from developing economies[END_REF]) that liberalization of financial markets leads to better redistribution of resources, a higher level of investment and higher efficiency of investment, and, therefore, has a positive impact on financial development. However, this influence weakened by the end of the period under study.

The ideas by Zingales [START_REF] Zingales | The Weak Links[END_REF], Huang and Temple [START_REF] Huang | Will political liberalisation bring about …financial development[END_REF][START_REF] Huang | Does external trade promote financial development?[END_REF] about the impact of openness on financial development were also confirmed on the sample of the EU countries in the period 1995 -2017: having examined the impact of the determinant of openness which is expressed by the Freedom to Trade Internationally index, a positive linear dependence was determined. However, the dependence significantly weakened by the end of the period under study.

The idea by La Porta, L´opez de Silanes (L´opez de Silanes et al. 1998) about the relations between the legal system and the degree of market development is confirmed: having examined the impact of the determinant of legal traditions expressed by the Legal System and Property Rights index on financial development, a strong positive linear dependence was determined during the whole period 1995 -2017.

The ideas by Voghouei [START_REF] Voghouei | A survey of the determinants of financial development[END_REF]), Zingales L., Rajan R. [START_REF] Zingales | Saving Capitalism from the Capitalists[END_REF] about the impact of political factors on financial development were confirmed: there is a positive linear dependence between the values of the financial development index and political stability index. However, the values of Pearson coefficients have a tendency to decrease in the period 1995 -2017. During this period, the linear dependence changes from being strong to being very weak: in 2007 the correlation coefficient comprised only 0.088.

Acemoglu's ideas (Acemoglu et al. 2005) about such a key factor of financial development as economic institutions were confirmed. The determinant characterizing economic institutions includes factors that determine the structure of incentives within society (i.e. incentives for economic agents to invest, and accumulate factors of productions, to make deals, etc.) and provide the redistribution of resources in the economy. The linear dependence of Financial Development index on the Freedom of Business index is positive and does not have a

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ISSN 2669-0195 (online) http://jssidoi.org/jesi/ 2020 Volume 2 Number 2 (June) http://doi.org/10.9770/IRD.2020.2.2(1) 519 vividly expressed tendency to change: the coefficients range from 0.562 to 0.314 in the period 1995 -2017. The linear dependence of Financial Development index on the Financial Freedom index is positive in the period 1995 -2017 (except for the year 2006), although it becomes weaker: from 0.552 to 0.236. However, during the years of recession there was a sharp decrease of the linear dependence, followed by an increase. The linear dependence of the Financial Development index on the Freedom of Investment index was positive in the period 1995 -2017. Despite some fluctuations, it tends to increase (in 1995, the value of correlation coefficient comprised 0.216; by 2017 it increased to 0.470).

The ideas by [START_REF] Huybens | Inflation, financial markets and long-run real activity[END_REF] about the negative impact of inflation on financial development were partially confirmed: the influence of inflation on financial development is characterized by a linear dependence. However, from 1995 to 2013 it was negative (the correlation coefficient ranges from -0.330 to -0.739), i.e. it had a negative impact on financial development. From 2014 to 2016 it was positive (the correlation coefficient ranges from 0.104 to 0.379); in 2017, it was negative again (the correlation coefficient comprised -0.402).

Levine's ideas [START_REF] Levine | Finance and growth: theory and evidence[END_REF] about the impact of investment in fixed capital on financial development in the EU countries are confirmed. There is an extremely weak linear dependence during both the pre-recession and postrecession period. The linear dependence is expressed by the average value of Pearson correlation coefficients during the recession and economic recovery periods.

The following conclusions can be drawn about the determinants of financial development of the EU countries in the period 1995-2017: 1) the impact of the determinants of openness, political stability characterizing the state of political rights and civil liberties, and financial liberalization with the increasing influence of government regulation on financial development of the EU countries in the period 1995 -2017 is decreasing significantly; 2) the impact of the determinant of legal traditions which includes judicial independence, impartial courts, protection of property rights, etc., as well as the impact of the determinant of economic institutions on financial development of the EU countries in the period 1995 -2017 remain very strong; 3) there are divergent trends (both enhancing and weakening) of the impact of the considered macroeconomic indices on financial development of the EU countries in the period 1995 -2017. It is obvious that the abovementioned determinants to a greater or lesser extent affect the financial development of the EU countries, but the degree of their influence varies during the period under study.

  INSIGHTS INTO REGIONAL DEVELOPMENT ISSN 2669-0195 (online) http://jssidoi.org/jesi/ 2020 Volume 2 Number 2 (June) http://doi.org/10.9770/IRD.2020.2.2(1) 510 2. Design and the sample of the research In the research, financial development is reflected by the indicators of financial development (Rethinking Financial Deepening: Stability and Growth in Emerging Markets) in the EU countries available in the period 1995 -2017. The indicator of financial development has the following structure (Figure 1):

Fig. 1 .

 1 Fig. 1. Structure of financial development indicator Source: Rethinking Financial Deepening: Stability and Growth in Emerging Markets. Imfstaffdiscussion note 2015. https://www.imf.org/external/pubs/ft/sdn/2015/sdn1508.pdf Note: FDfinancial development; FIfinancial institutions; FMfinancial markets;FID -financial institutions depth; FIA -financial institutions access; FIE -financial institutions efficiency; FMD -financial markets depth; FMA -financial markets access; FME -financial markets efficiency. FID -financial institutions depth is characterized by the following indicators: private-sector credit (% of GDP), pension fund assets (% of GDP), mutual fund assets (% of GDP), insurance premiums, life and non-life (% of GDP). FIA -financial institutions access is characterized by the following indicators: branches (commercial banks) per 100,000 adults; ATMs per 100,000 adults. FIE -financial institutions efficiency is characterized by the following indicators: net interest margin; lending-deposits spread; non-interest income to total income; overhead costs to total assets; return on assets; return on equity. FMD -financial markets depth is characterized by the following indicators: stock market capitalization to GDP; stocks traded to GDP; international debt securities government (% of GDP); total debt securities of nonfinancial corporations (% of GDP); total debt securities of financial corporations (% of GDP). FMA -financial markets access is characterized by the following indicators: percent of market capitalization outside of top 10 largest companies; total number of issuers of debt (domestic and external, nonfinancial corporations, and financial corporations). FME -financial markets efficiency is characterized by the following indicator: stock market turnover ratio (stocks traded/capitalization).

Fig. 2 .

 2 Fig. 2. Pearson correlation coefficients which characterize the linear dependence of the Financial Development index on the Economic Freedom index (р-value<0.01). Source: the authors' calculations on the data from Rethinking Financial Deepening

Fig. 11 .

 11 Fig. 11. Pearson correlation coefficients in the period 1995 -2017 between the values of the Financial Development index and Gross Fixed Capital Formation (% of GDP) (р-value<0.01). Source: the authors' calculation on the data from Rethinking Financial Deepening

  studied economic institutions in more detail dividing them into the following groups:

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Table 1 .

 1 Determinants of financial development: empirical interpretation of indexes

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Volume 2 Number 2 (June) http://doi.org/10.9770/IRD.2020.2.2(1)

Acknowledgements

This research was partly supported by European Union's Horizon 2020 Research and Innovation Programme European Cohort Development Project (ECDP) Grant Agreement

This research was partly supported by European Union's Horizon 2020 Research and Innovation Programme European Cohort Development Project (ECDP) Grant Agreement

Regulation index: business regulations, labour market regulations, credit market regulations (р-value<0.01).

Source: the authors' calculations on the data from Rethinking Financial Deepening

There is a positive impact of the determinant of financial liberalization on financial development in the period under study in the EU countries, but this influence decreases sharply every year (Figure 3). However, the impact of government regulation on financial development becomes stronger. For example, J. Stiglitz [START_REF] Stiglitz | Capital Market Liberalization, Economic Growth and Instability[END_REF] believed that strategies which included financial liberalization (as well as liberalization of capital market) did not take into account one important factor: capital flows are pro-cyclical, so claims that opening of capital markets encourages diversification and increases stability were incomplete. This led to the conclusion that it was necessary to review permitted ways for government intervention which could stabilize the short-term movement of capital, and, therefore, he considered government regulation to be justified.

Freedom to Trade Internationally index reflects international trade taxes (revenue from trade taxes as a share of export and import, mean tariff rate, standard deviation of tariff rates) and regulatory trade barriers (non-tariff trade barriers, compliance costs of importing and exporting), as well as actual volume of foreign trade as compared to the expected one, difference between official exchange rates and black-market exchange rates, government control on international movement of capital (foreign ownership and investment restrictions, restrictions of freedom for citizens to participate in capital operations with foreign partnersthe index of monitoring capital operations according to 13 categories of International Monetary Fund).

Having examined the influence of the determinant of openness, expressed by the Freedom to Trade Internationally index on financial development, a positive linear dependence is determined. However, the value of the Pearson correlation coefficients is gradually decreasing from 0.632 in 1995 to 0.062 in 2017 (Figure 4). Financial development can be neither provided nor be successful if the ruling elite opposes it. Zingales and Rajan and explain that the poor development of financial markets creates an environment for officials of large and wellknown industrial or financial companies to benefit from monopolistic nature of rents. Companies' market power under a poor disclosure system and weak contracts imposes high entry barriers for new potential agents. Therefore, the management of such companies (having a strong influence in the government) often opposes the openness and financial development. "Open borders decrease the opportunities for politicians to decrease the level of competition and slow down financial and economic growth" [START_REF] Zingales | Saving Capitalism from the Capitalists[END_REF].

Having examined the impact of the determinant of legal traditions expressed by the Legal System and Property Rights index on financial development, a positive linear dependence is determined. The Legal System and Property Rights index reflexes judicial independence, impartial courts, protection of property rights, military interference in rule of law and politics, integrity of the legal system, legal enforcement of contracts, regulatory costs of the sale of real property, reliability of police, and business costs of crime. Protection of persons and their rightfully acquired property is a central element of civil society. Values of Pearson correlation coefficients decreased insignificantly from 1995 to 2017, and comprises 0.716 and 0.562 respectively. Legal system which guarantee and provide property rights is a key basis for financial development, which was confirmed in the study conducted by L. Zingales and R. Rajan [START_REF] Zingales | Saving Capitalism from the Capitalists[END_REF]. See Figure 5. The determinant characterizing Political Power and Political Institutions is expressed by the Political Stability index which represents the data of the survey on 2 areas: political rights (0-40 points) and civil liberties (0-60 points) (Freedom in the World is Freedom House's flagship annual report assessing the condition of political rights and civil liberties around the world). Free press, active public participation in political life and competitive political parties contribute to the solution of the issue on pressure imposed by influential persons who are interested in the development of only those institutions which are beneficial to them and which will support their power. However, it possible to effectively prevent the subordination of economic policy to the interests of new influential circles only in the context of foreign economic competition... which forces politicians to pursue a more effective policy aimed at the market development [START_REF] Zingales | Saving Capitalism from the Capitalists[END_REF]. There is a positive linear dependence between the values of the Financial Development index and Political Stability index. However, there is a trend for decline in the values of Pearson coefficients in the period 1995 -2017. During this period, the linear dependence changed from being strong to very weak; it comprises 0.088 in 2007. See Figure 6. In the period 1995 -2017, the linear dependence between the Financial Development index and Economic Freedom index is positive (except for 2006), but it becomes weaker from 0.552 to 0.236. However, during the years of economic recession, the linear dependence sharply decreases, and then increases. (See Figure 8). Macroeconomic determinants: inflation, gross fixed capital formation. A linear dependence characterizes the impact of inflation on financial development. However, in the period 1995 -2013 it was negative (the correlation coefficient ranges from -0.330 to -0.739), i.e. it had a negative impact on financial development. In the period 2014 -2016, it was positive (the correlation coefficient ranges from 0.104 to 0.379). In 2017, it again became negative (the correlation coefficient -0.402). The higher the inflation is, the lower the return on money is, which leads to decrease in credit activities. If a financial sector provides less credit, it means that the distribution of money is inefficient, and has a negative effect on financial development (See Figure 10).