Xue Luo

Dong Ye
email: dong.ye@univ-metz.fr

Feng Zhou
email: fzhou@math.ecnu.edu.cn

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

Keywords: singular nonlinearity, advection, extremal solution, regularity * 35B65, 35B45, 35J60

In this note, we investigate the regularity of the extremal solution u * for the semilinear elliptic equation -u + c(x) • ∇u = λf (u) on a bounded smooth domain of R n with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a ∈ (0, ∞). We show that the extremal solution is regular in the low dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.

Introduction

We consider the elliptic problem

     -u + c(x) • ∇u = λf (u) in Ω, u > 0
in Ω, u = 0 on ∂Ω,

(P λ )
where λ > 0, Ω is a smooth bounded domain in R n (n ≥ 2), c(x) is a smooth vector field over Ω and f : [0, a) → R + with fixed a ∈ (0, ∞) satisfies the following condition (H):

f is C 2 , positive, nondecreasing and convex in [0, a) with lim

t→a - f (t) = ∞.
In the literature, f is refered as a singular nonlinearity. We say that u is a regular solution if u ∈ C 2 (Ω), and we also deal with solutions in the following weak sense.

Definition 1.1. We say that u is a weak solution of (P λ ) if 0 ≤ u ≤ a a.e. in Ω such that f (u)d(x, ∂Ω) ∈ L 1 (Ω) and

- Ω u∆φ - Ω udiv(φc) = λ Ω f (u)φ, ∀ φ ∈ C 2 (Ω) ∩ H 1 0 (Ω).
Moreover, u is a weak super-solution of (P λ ) if " = " is replaced by " ≥ " for all nonnegative functions φ ∈ C 2 (Ω) ∩ H 1 0 (Ω).

Clearly, a weak solution is regular if sup Ω u < a. For regular solutions, we introduce a notion of stability. Definition 1.2. A regular solution u of (P λ ) is said to be stable if the principal eigenvalue of the linearized operator L u,λ,c := -+ c • ∇ -λf (u) is nonnegative in H 1 0 (Ω).

Exploiting some ideas in [START_REF] Martel | Uniqueness of weak extremal solutions of nonlinear elliptic problems[END_REF][START_REF] Ghoussoub | On the partial differential equations of electro MEMS devices: stationary case[END_REF][START_REF] Brezis | Blow-up solutions of some nonlinear elliptic problems[END_REF], the solvability of (P λ ) is characterized by a parameter λ * : Proposition 1.1. There exists λ * ∈ (0, ∞) such that • For 0 < λ < λ * , the problem (P λ ) has a minimal solution u λ , u λ is regular and the map λ → u λ is increasing. Moreover, u λ is the unique stable solution of (P λ ).

• For λ = λ * , (P λ * ) admits a unique weak solution u * := lim λ→λ * u λ , called the extremal solution.

• For λ > λ * , (P λ ) admits no weak solution.

Here the minimal solution means that u λ ≤ v for any solution v of (P λ ). We remark immediately a close similarity between (P λ ) and the Emden-Fowler equation with superlinear regular nonlinearity, that is

-∆u = λg(u) in Ω ⊂ R n ; u = 0 on ∂Ω, (1.1)
with λ > 0 and g : [0, ∞) → (0, ∞) satisfies g is C 2 , nondecreasing, convex and lim

t→∞ g(t) t = ∞. (1.2)
In fact, there exists also a critical parameter λ ∈ (0, ∞) for (1.1) such that all conclusions in the above proposition hold true by replacing λ * by λ (see [START_REF] Brezis | Blow up for u t -∆u = g(u) revisited[END_REF][START_REF] Martel | Uniqueness of weak extremal solutions of nonlinear elliptic problems[END_REF]). It is well known by classical examples as g(u) = (1 + u) p with p > 1 or g(u) = e u , the extremal solution u * can be either a regular solution or a real weak solution in the distribution sense with sup Ω u = ∞.

For general nonlinearity g satisfying (1.2), the regularity of the extremal solution u * to (1.1) is obtained by Nedev [START_REF] Nedev | Regularity of the extremal solution of semilinear elliptic equations[END_REF] for any bounded smooth domain Ω ⊂ R n if n = 2, 3; by Cabré [START_REF] Cabré | Regularity of minimizers of semilinear elliptic problems up to dimension 4[END_REF] for convex domains in R 4 ; and for radial symmetry case in R n with n ≤ 9 by Cabré & Capella [START_REF] Cabré | Regularity of radial minimizers and extremal solutions of semilinear elliptic equations[END_REF]. In [START_REF] Ye | Boundedness of the extremal solution for semilinear elliptic problems[END_REF], it is proved that, under mild condition on g, the extremal solution u * is regular for any smooth bounded domain Ω ⊂ R n if n ≤ 9.

We can ask the same question about the problem (P λ ): For f verifying (H), is it true that the extremal solution to (P λ ) is regular for general vector field c and general domain Ω ⊂ R n with low dimensions n? We will partly answer this question. It is worthy to mention that for studying the explosion phenomena in a flow, Berestycki et al. [START_REF] Berestycki | The explosion problem in a flow[END_REF] have considered the problem (P λ ) with a general source f verifying (1.2).

Without loss of generality, fix a = 1 in the sequel. The problem (P λ ) can be linked to equation (1.1) up to the transformation v = -ln (1 -u). In fact, let u solve (P λ ), v verifies then

-v + |∇v| 2 + c(x) • ∇v = λe v f (1 -e -v ) := λg(v) in Ω, v = 0 on ∂Ω. (Q λ )
Therefore g verifies (1.2) and v * = -ln(1 -u * ) is the extremal solution for the problem (Q λ ). Thus the regularity of u * is equivalent to the boundedness of v * , however the situation could be very different with the presence of advection terms (see [START_REF] Cowan | Regularity of the extremal solution in a MEMS model with advection[END_REF][START_REF] Wei | On MEMS equation with fringing field[END_REF]). In last decade, a model describing the steady state of MEMS (Micro-Electro-Mechanical Systems) device given by Pelesko and Bernstein in [START_REF] Pelesko | Modeling MEMS and NEMS[END_REF], has drawn many attentions (see [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] and the references therein).

-∆u = λ (1 -u) 2 in Ω ⊂ R n ; u = 0 on ∂Ω.
More generally, many precise studies have been done for the singular nonlinearities with negative exponent f (u) = (1 -u) -p (p > 0) in the advection-free situation, i.e. c ≡ 0. In that case, when Ω is moreover the unit ball in R n , it is known that u * is regular if and only if (see [START_REF] Mignot | Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe[END_REF][START_REF] Ghoussoub | On the partial differential equations of electro MEMS devices: stationary case[END_REF])

n < n p := 2 + 4p p + 1 + 4 p p + 1 . (1.3)
Tending p → 0 + in (1.3), we see that n p → 2. Therefore we cannot expect in general better than dimension two to claim the regularity of u * .

For the radial case of (P λ ), equally when Ω is a ball and c(x) is the gradient of a smooth radial function, u λ is radial by uniqueness of the minimal solution. We obtain the following optimal results which are new even for the advection-free case.

Theorem 1.1. Assume that n = 2, Ω = B 1 . Let γ is a smooth radial function and c = ∇γ, then the extremal solution u * is regular for any f satisfying (H).

Theorem 1.2. For any f satisfying (H), Ω = B 1 and smooth radial function γ, there exists C > 0 such that for all λ ∈ (0, λ * ]

|u λ (r)| ≤ Cr -1 if n ≥ 10; Cr -n 2 +1+ √ n-1 if 3 ≤ n ≤ 9; ∀ r = |x| ∈ (0, 1]
where

| • | is the Euclidean norm in R n .
Remark 1.1. The above estimates are optimal. In fact, when f (u) = (1 -u) -p , p > 0, Ω = B 1 and c ≡ 0, it is well known that u * (x) = 1 -r 2 p+1 if n ≥ n p with n p given in (1.3), and we have

n ≥ n p iff n ≥ 10 or 3 ≤ n ≤ 9, 2 p + 1 ≤ - n 2 + 2 + √ n -1.
But is the extremal solution u * of (P λ ) regular with general singular nonlinearity f verifying (H), vector field c and smooth bounded domains in R 2 ? The answer is affirmative under some additional mild condition on f . Theorem 1.3. Assume that f satisfies conditions (H) and the additional conditions, (H1) lim sup

t→1 - f (t) f (t)(1 -t) ln 2 (1 -t) < 1
and

(H2) lim inf t→1 - f (t)f (t) f 2 (t) > 0.
Then u * is regular solution to

(P λ * ) if n = 2, i.e. Ω ⊂ R 2 .
Under more precise conditions on the growth of f , the extremal solution can be showed to be regular in some higher dimensions.

Theorem 1.4. Let f verify (H) and g(v) = e v f (1 -e -v ). Assume that g satisfies

(H3) lim inf t→∞ g (t) g(t) = 1 + δ > 1 and ( H2) lim inf t→∞ g (t)g(t) g 2 (t) = µ > 1 1 + δ . Then v * = -ln(1 -u * ) is bounded (so u * is regular) when n < 2 + 4δ 1 + δ + 4 δ(µ + µδ -1) 1 + δ . (1.4)
Consequently, if µδ > 1, u * is regular for all n ≤ 6. Furthermore, if we can tend δ to ∞, which means g = o(g ) near ∞, then u * is regular for n < 6 + 4

√ µ with any µ > 0. However, we can never have µ > 1, since otherwise g blows up at finite value and contradicts (1.2), so the best result we can expect is for n ≤ 9. For example, if t) in (0, ∞), suppose there exists t 0 > 0 such that t 2 h (t) is nondecreasing for t ≥ t 0 , then for any bounded smooth domain Ω ⊂ R n with n ≤ 9, u * is a regular solution.

f (u) = e 1 1-u , then g(v) = e v+e v verifies δ = ∞ and µ = 1. Theorem 1.5. Let f verify (H) and g(v) = e v f (1 -e -v ). Assume that g = o(g ) near ∞. Rewrite g(t) = g(0) + te h(
Furthermore, when g = o(g ) near ∞, the condition ( H2) is just equivalent to (H2), since

f (t)f (t) f 2 (t) = (g -g ) g (g -g) 2 (s) = g g g 2 - g g × 1 - g g -2 (s), ∀ t = 1 -e -s .
It is also easy to see that (H3) is equivalent to the condition lim inf

t→1 - f (t)(1 -t) f (t) = δ > 0.
If the equality holds for the whole limit, we have the following optimal result. The case f (u) = (1 -u) -2 was obtained in [START_REF] Cowan | Regularity of the extremal solution in a MEMS model with advection[END_REF] with a different argument.

Theorem 1.6. Assume that

lim u→1 - f (u)(1 -u) f (u) = p > 0. (1.5)
Then u * is a regular solution if n < n p where n p is defined in (1.3).

One of the main difficulties here is due to the vector field c(x). When c = 0, the operator -∆ + c • ∇ is not self-adjoint, we use ideas from [START_REF] Cowan | Regularity of the extremal solution in a MEMS model with advection[END_REF] to get some energy estimates. However if c is a gradient, say c = -∇γ in Ω, then -∆ + c • ∇ can be rewritten as e -γ L γ where L γ = -div(e γ ∇) is a self-adjoint operator. In that case, (P λ ) admits a variational structure and we can expect more precise estimates of minimal solutions u λ , as in the radial case.

The paper is organized as follows: In section 2, we prove quickly Proposition 1.1 and show some general consequences of the stability of u λ . The section 3 is devoted to the proof of Theorems 1.3 to 1.6 for general domains. In section 4, we discuss the radial case. The norm • q denotes always the standard L q norm for any q ∈ [1, ∞]. The capital letter C denotes a generic positive constant independent of λ, it could be changed from one line to another.

Preliminaries

As mentioned above, -∆ + c • ∇ is not a self-adjoint operator for general vector field c. However using Lemma 1 in [START_REF] Cowan | Regularity of the extremal solution in a MEMS model with advection[END_REF], we have a kind of Hodge decomposition, which tells us that for any vector field c ∈ C ∞ (Ω, R n ), there exist a smooth scalar function γ and a

vector field b ∈ C ∞ (Ω, R n ) such that c = -∇γ + b and div(e γ b) = 0 in Ω. (2.1)
Therefore the problem (P λ ) can be rewritten as

-div(e γ ∇u) + e γ b • ∇u = λe γ f (u) in Ω. (P λ )
On the other hand, we don't have a suitable variational characterization in general to use the stability assumption. Fortunately, we can adopt an energy inequality as in [START_REF] Cowan | Regularity of the extremal solution in a MEMS model with advection[END_REF], which is derived from a generalized Hardy inequality of [START_REF] Cowan | Optimal Hardy inequalities for general elliptic operators with improvemnets[END_REF].

Proposition 2.2. Let u λ be minimal solution of (P λ ). For any 1 ≤ β < 2, we have

λ Ω e γ f (u λ )ψ 2 ≤ 2 β Ω e γ |∇ψ| 2 + b 2 ∞ 2(2 -β) Ω e γ ψ 2 , ∀ ψ ∈ H 1 0 (Ω). (2.2)
where b is the vector field in

(2.1), b ∞ = max Ω |b(x)|.
Proof. We use a Hardy type inequality given by Theorem 2 in [START_REF] Cowan | Regularity of the extremal solution in a MEMS model with advection[END_REF], which says that for a positive principal eigenfunction

ϕ of L u λ ,λ,c , for β ∈ [1, 2) and any ψ ∈ H 1 0 (Ω), λ Ω e γ f (u λ )ψ 2 ≤ 2 β Ω e γ |∇ψ| 2 + Ω - 2 -β 2
|∇ϕ| 2 ϕ 2 + b • ∇ϕ ϕ e γ ψ 2 .
By Cauchy-Schwarz inequality, it is easy to see

- 2 -β 2
|∇ϕ| 2 ϕ 2 + b • ∇ϕ ϕ ≤ |b(x)| 2 2(2 -β) ≤ b 2 ∞ 2(2 -β) ,
so we are done.

Another main ingredient of our approach is just the transformation v = -ln(1 -u). Let φ and ξ be nonnegative

C 1 functions satisfying φ(0) = ξ(0) = 0 and ξ = φ 2 . Define v λ = -ln(1 -u λ ) and g(v λ ) = e v λ f (1 -e -v λ ). Using (Q λ ), we get -div(e γ ∇v λ ) + e γ b • ∇v λ ≤ λe γ g(v λ ) in Ω. Let ψ = φ(v λ ) in (2.2), ∀ λ ∈ (0, λ * ), λ Ω e γ f (u λ )φ 2 (v λ ) ≤ 2 β Ω e γ |∇φ(v λ )| 2 + b 2 ∞ 2(2 -β) Ω e γ φ 2 (v λ ) = 2 β Ω e γ ∇ξ(v λ )∇v λ + C β Ω e γ φ 2 (v λ ) = - 2 β Ω div(e γ ∇v λ )ξ(v λ ) + C β Ω e γ φ 2 (v λ ) ≤ 2λ β Ω e γ g(v λ )ξ(v λ ) - 2 β Ω e γ b • ξ(v λ )∇v λ + C β Ω e γ φ 2 (v λ ) = 2λ β Ω e γ g(v λ )ξ(v λ ) + C β Ω e γ φ 2 (v λ ).
The last line is due to div(e γ b) = 0. We claim then

Proposition 2.3. Let 1 ≤ β < 2.
For any λ ∈ (0, λ * ) and any nonnegative C 1 test functions φ, ξ verifying φ(0) = ξ(0) = 0 and ξ = φ 2 , there hold

λ Ω e γ f (u λ )φ 2 (v λ ) ≤ 2λ β Ω e γ g(v λ )ξ(v λ ) + C β Ω e γ φ 2 (v λ ) (2.3) and λ Ω e γ f (u λ )φ 2 (u λ ) ≤ 2λ β Ω e γ f (u λ )ξ(u λ ) + C β Ω e γ φ 2 (u λ ). (2.4)
The proof of (2.4) is completely similar to (2.3) but using (P λ ) instead of (Q λ ).

We also make use the following behavior of f proved in [START_REF] Ye | On a general family of nonautonomous elliptic and parabolic equations[END_REF].

Lemma 2.1. For any f verifying (H), we have lim t→1 f (t)/f (t) = 0.

Choose first φ(u) = e u -1 in (2.4), then ξ(u) = e 2u -1

2 and λ Ω e γ f (u λ ) (e u λ -1) 2 ≤ λ β Ω e γ f (u λ ) e 2u λ -1 + C β Ω e γ (e u λ -1) 2 .
Fix β ∈ (1, 2). By Lemma 2.1,

λ Ω e γ f (u λ )e 2u λ ≤ C. Consequently f (u λ ) 1 is uniformly bounded, so is f (u λ ) 1 . Multiplying (P λ ) by u λ , Ω |∇u λ | 2 = Ω div(c) 2 u 2 λ + λ Ω f (u λ )u λ ≤ C,

which gives

Proposition 2.4. The family of minimal solutions {u λ } 0<λ<λ * is uniformly bounded in H 1 0 (Ω).

Remark 2.1. As far as we know, it is always an open question whether the similar H 1 energy estimation holds for minimal solutions of (1.1) with general regular nonlinearity satisfying (1.2) and general domain Ω when n ≥ 6 (see [START_REF] Nedev | Regularity of the extremal solution of semilinear elliptic equations[END_REF] for n ≤ 5). For the advectionfree case c = 0, it was proved in [START_REF] Ye | On a general family of nonautonomous elliptic and parabolic equations[END_REF] that u * ∈ H 2 ∩ H 1 0 (Ω) under the condition (H), it is also true for the gradient case c = ∇γ (see Lemma 4.1).

Sketches of proof of Proposition 1.1. We follow the ideas coming from [START_REF] Berestycki | The explosion problem in a flow[END_REF][START_REF] Martel | Uniqueness of weak extremal solutions of nonlinear elliptic problems[END_REF][START_REF] Ghoussoub | On the partial differential equations of electro MEMS devices: stationary case[END_REF]. The main argument is the maximum principle for operators -∆ + c • ∇ and L γ under the Dirichlet boundary condition, we use also the super-sub solution method and monotone iteration.

Let w ∈ H 1 0 (Ω) be the regular solution of -∆w + c • ∇w = 1 in Ω and fix α > 0 such that α max Ω w < 1. It is easy to verify that αw is a supersolution of (P λ ) for λ > 0 small enough. As 0 is a subsolution and αw > 0 in Ω, (P λ ) admits a regular solution for λ > 0 small enough. As any regular solution u of (P λ ) is also a supersolution for (P µ ) if µ ∈ (0, λ), the set of λ for which (P λ ) admits a regular solution is just an interval. Moreover, for these λ, using (H) and the monotone iteration

v 0 = 0; -∆v n+1 + c • ∇v n+1 = λf (v n )
in Ω with v n+1 = 0 on ∂Ω for n ∈ N, we get the minimal solution u λ = lim n→∞ v n .

If we suppose that the principal eigenvalue of L u λ ,λ,c is negative, we can construct, as in [START_REF] Berestycki | The explosion problem in a flow[END_REF] another solution v ≤ u λ using the associated first eigenfunction, this is just impossible by the definition of u λ , hence u λ is stable. The uniqueness of stable solution comes from Lemmas 2.16 and 2.17 in [START_REF] Crandall | Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems[END_REF]. Take a positive first eigenfunction ϕ of L γ with the Dirichlet boundary condition, by (P λ ),

λf (0) Ω e γ ϕ ≤ Ω λe γ f (u)ϕ = Ω λ 1 (L γ )uϕ - Ω div(e γ bϕ)u ≤ C.
So λ is upper bounded. Define the critical threshold λ * as the supermum of λ > 0 for which (P λ ) admits a regular solution, as u * is the monotone limit of u λ when λ → λ * , we deduce that u * ∈ H 1 0 (Ω) is a weak solution of (P λ ) by Proposition 2.4. Suppose that u is a weak solution to (P λ ). By the monotonicity of f , it is easy to verify that for any δ > 1, the function v = δ -1 u is a weak supersolution for (P λ/δ ), then the monotone iteration will enable us a weak solution w of (P λ/δ ) satisfying 0 ≤ w ≤ v ≤ δ -1 < 1. The regularity theory implies then w is a regular solution of (P λ/δ ). This means that λ/δ ≤ λ * . Let δ tend to 1, we get λ ≤ λ * . Therefore, no weak solution exists for λ > λ * .

The uniqueness of the weak solution can be proved in the very similar way as in [START_REF] Martel | Uniqueness of weak extremal solutions of nonlinear elliptic problems[END_REF] using the monotonicity and convexity of f , with the strong maximum principle for the operator -∆ + c • ∇ associated to Dirichlet boundary condition, so we omit the details.

Regularity of u * for general c and Ω

For proving our results, we will choose suitable functions φ to apply (2.3) or (2.4). We need also Lemma 3.1. For any q > n/2, there exists C > 0 such that the solution v of (Q λ ) satisfies 0 ≤ v ≤ C g(v) q in Ω.

Indeed, let w be the solution of L(w) := -∆w + c • ∇w = λg(v) in Ω with w = 0 on ∂Ω. By regularity theory and Sobolev embedding, w ∞ ≤ C w W 2,q (Ω) ≤ C λ * g(v) q because q > n/2 ≥ 1. Morover, as L(w -v) ≥ 0, the maximum principle implies then 0 ≤ v ≤ w ≤ C g(v) q .

Proof of Theorem 1.3

For simplicity, we omit the index λ for u λ or v λ . Let φ(u) = v = -ln(1-u) in (2.4), so ξ(u) = (1 -u) -1 -1. Fix β ∈ (1, 2) but very close to 2. Repeating the proof of Theorem 2 in [START_REF] Ye | On a general family of nonautonomous elliptic and parabolic equations[END_REF] with the assumption (H1), there exists C > 0 such that

λ Ω e γ f (u) 1 -u < C + CC β Ω e γ φ 2 (u). As φ 2 (u) = o(ξ(u)) = o(f ξ) when u → 1 -, λ Ω e γ f (u) 1 -u ≤ C.
Using the equation (Q λ ) and

∂ ν v ≤ 0 on ∂Ω, Ω |∇v| 2 = λ e v f (1 -e -v ) + ∂Ω ∂v ∂ν dσ - Ω c • ∇v ≤ λ Ω f (u) 1 -u + C ∇v 2 ≤ C + C ∇v 2 .
Therefore ∇v 2 ≤ C, the classical Moser-Trudinger inequality enables us, as n = 2 Ω e qv ≤ C q , ∀ q ≥ 1.

(3.1)

Take now φ(u) = f (u) -f (0) in (2.4), we need to estimate

ζ(u) := f (u)φ(u) - 2 β ξ(u) = f (u)φ(u) - 2 β u 0 f 2 (s)ds = f (u)f (u) - 2 β u 0 f 2 (s)ds -Cf (u) := I(u) - 2 β J(u) -Cf (u).
By (H2), there exists δ > 0 such that

I(u) -I(0) = u 0 f 2 (s) + f (s)f (s) ds ≥ (1 + δ)J(u) -Cf (u), ∀ u ∈ [0, 1) Let 4 2+δ < β < 2, we get ζ(u) ≥ CI(u) -C. Asserting this in (2.4), λ Ω e γ f (u)f 2 (u) ≤ C Ω e γ f 2 (u) + C.
Consequently, f (u)f 2 (u) 1 ≤ C. By Lemma 2.1, we deduce f (u) 3 ≤ C. Combining with (3.1), g(v) p ≤ C for any p < 3. The proof is completed by Lemma 3.1 as n = 2.

Proof of Theorem 1.4

Without loss of generality, we can assume that g(0) = 1. Let φ(t) = g α (t) -1 where α > 0 is a constant to be determined later. Then

ξ(t) = t 0 φ 2 (s)ds = α 2 t 0 g 2α-2 (s)g 2 (s)ds = α 2 2α -1 g 2α-1 (t)g (t) - α 2 2α -1 t 0 g 2α-1 (s)g (s)ds -C α . (3.2)
The condition ( H2) yields: Given any ∈ 0, µ -1 1+δ , there exists

C ≥ 0 such that g(t)g (t) ≥ (µ -)g 2 (t) -C in [0, ∞). Therefore - t 0 g 2α-1 (s)g (s)ds ≤ -(µ -) t 0 g 2α-2 (s)g 2 (s)ds + C ≤ - µ - α 2 ξ(t) + C. (3.3)
We divide the proof into two cases.

Case 1: δ > 1 and µ > 1 1+δ ; or δ ≤ 1 with µ > 1+δ 4δ . Take α > 1 2 . Combine (3.2) and (3.3),

1 + µ - 2α -1 ξ(t) ≤ α 2 2α -1 g 2α-1 (t)g (t) + C, consequently ξ(t) ≤ α 2 2α -1 + µ - g 2α-1 (t)g (t) + C, for any t ≥ 0. (3.4)
According to (H3), for any 0 < δ < δ, there exists C > 0 such that g (t) ≥ (1+δ )g(t)-C in [0, ∞). Setting these estimates in (2.3), omitting the index λ and recalling that f

(u) = g (v) -g(v), δ λ 1 + δ Ω e γ g (v)(g α (v) -1) 2 -Cλ Ω e γ (g α (v) -1) 2 ≤ λ Ω e γ f (u)(g α (v) -1) 2 ≤ 2α 2 λ β(2α -1 + µ -) Ω e γ g 2α (v)g (v) + Cλ Ω e γ g(v) + C Ω e γ (g α (v) -1) 2 .
Consequently,

δ 1 + δ - 2α 2 β(2α -1 + µ -) λ Ω e γ g (v)g 2α (v) ≤ 2δ C 1 + δ Ω e γ g (v)g α (v) + C Ω e γ g(v) + C Ω e γ (g α (v) -1) 2 .
Choose δ near δ such that either δ > 1 and µ >

1 1 + δ or δ < δ ≤ 1 with µ > 1 + δ 4δ .
Through direct computations, for > 0 sufficiently small and β = 2 -, there exists

α ∈ 1 2 , δ 1 + δ + δ (1 + δ )(µ -) -δ 1 + δ such that δ 1 + δ - 2α 2 β(2α -1 + µ -) > 0. ( 3.5)
For such α, we obtain

λ Ω e γ g 2α (v)g (v) ≤ C, ∀ λ ∈ (0, λ * ). (3.6)
Tending now δ to δ and to 0, (3.6) holds true provided that

α < δ 1 + δ + δµ(1 + δ) -δ 1 + δ . (3.7)
Therefore

Ω e γ g 2α+1 (v) ≤ C Ω e γ g 2α (v)g (v) + C ≤ C,
which implies that g(v) 2α+1 ≤ C for α verifying (3.7). Applying Lemma 3.1, we conclude that for n < 2 + 4α with α verifying (3.7), v λ is uniformly bounded, hence u * is a regular solution if n satisfies (1.4).

Case 2: δ ≤ 1 and 1 1+δ < µ ≤ 1+δ 4δ . Now we take α ∈ 1 2 (1 -µ + ), 1 2 , the formulas (3.2) and (3.3) imply then

1 + µ - 2α -1 ξ(t) ≥ α 2 2α -1 g 2α-1 (t)g (t) + C.
The inequality (3.4) still holds true. Proceeding as for Case 1, we see that for δ < δ but nearby, > 0 small and β = 2 -, there exists

α ∈ 1 -µ + 2 , δ 1 + δ + δ (1 + δ )(µ -) -δ 1 + δ ⊂ 1 -µ + 2 , 1 2
such that (3.5) is satisfied. Hence we conclude exactly as in Case 1.

Proof of Theorem 1.5

Without loss of generality, assume again g(0) = 1. Take now φ(t) = te αh(t) , where α > 0 is a constant to be determined, then

ξ(t) = t 0 [1 + sαh (s)]
2 e 2αh(s) ds

= t 0 [1 + 2sαh (s)] e 2αh(s) ds + t 0 α 2 s 2 h 2 (s)e 2αh(s) ds = te 2αh(t) + K(t).
Thus, for t ≥ t 0 ,

2K(t) α = 2α t 0 s 2 h 2 (s)e 2αh(s) ds = C + t t0 s 2 h (s)d e 2αh(s) ≤ C + t 2 h (t)e 2αh(t) - t t0 e 2αh(s) d s 2 h (s) ,
where the last integration is considered in the sense of Stieltjes. The monotonicity of

s 2 h in [t 0 , ∞) yields K(t) ≤ α 2 t 2 h (t)e 2αh(t) + C, ∀ t ≥ t 0 .
So we get

ξ(t) ≤ C + t + α 2 t 2 h (t) e 2αh(t) , ∀ t ≥ 0.
Using (2.3) (we drop the index λ), v) , By Young's inequality,

Ω e γ e h(v) + vh (v)e h(v) -ve h(v) -1 v 2 e 2αh(v) ≤ 2 β Ω e γ 1 + ve h(v) ξ(v) + C Ω e γ v 2 e 2αh(v) ≤ 2 β Ω e γ 1 + ve h(v) C + ve 2αh(v) + α 2 v 2 h (v)e 2αh(v) + C Ω e γ v 2 e 2αh(
1 - α β Ω e γ v 3 h (v)e (2α+1)h(v) ≤ C Ω e γ 1 + v 2 h (v)e 2αh(v) + v 3 e (2α+1)h(v) . (3.8)
Moreover, g = o(g ) at infinity yields lim t→∞ h (t) = ∞, hence

t 2 h (t)e 2αh(t) + t 3 e (2α+1)h(t) t 3 h (t)e (2α+1)h(t) = 1 g(t) -1 + 1 h (t) → 0 as t → ∞.
Fix β ∈ (α, 2), the inequality (3.8) implies

Ω [g(v) -1] 2α+1 v 2α = Ω ve (2α+1)h(v) ≤ C + Ω v 3 h (v)e (2α+1)h(v) ≤ C.
Recall that g is superlinear, we obtain g(v) 1 ≤ C. Consider again w satisfying L(w) = λg(v) in Ω and w = 0 on ∂Ω, as v ≤ w in Ω by maximum principle,

Ω (g(v) -1) 2α+1 w 2α ≤ C.
Following the proof of Lemma 2.1 in [START_REF] Ye | Boundedness of the extremal solution for semilinear elliptic problems[END_REF] (we just need a minor adjustment, say define Ω 1 = {x ∈ Ω : g(v) > w T } instead, here T > 0 is a suitable constant), we can obtain that if 2α + 1 > n/2, w is uniformly bounded in L ∞ (Ω), so does v. Taking 2 > β > α > 7/4, the result holds for n ≤ 9.

Proof of Theorem 1.6

Here we choose φ(u) = (1 -u) -α -1 in (2.4). For 2λ > λ * and > 0, [START_REF] Berestycki | The explosion problem in a flow[END_REF] by (1.5). As > 0 is arbitrary,

p - 2α 2 β(2α + 1) -2 Ω e γ (1 -u) p+2α+1 ≤ C, ∀ β ∈ [1, 2). We have used f (u)(1 -u) ≥ (p -)f (u) -C in [0,
Ω 1 (1 -u) p+2α+1 ≤ C provided that p > α 2 2α + 1
, i.e. when α < p + p(p + 1).

Therefore (1 -u) -1 q ≤ C if q < 1 + 3p + 2 p(p + 1). For any > 0, as f (u)(1 -u) ≤ (p + )f (u) + C in [0, 1) by (1.5), we have f (u) ≤ C(1 -u) -p-, consequently

g(v) = e v f (1 -e -v ) = f (u) 1 -u ≤ C(1 -u) -1-p-, hence g(v) r ≤ C when r < 1 + 3p + 2 p(p + 1) p + 1 + .
According to Lemma 3.1, the proof is done by taking → 0 + .

Radial case

As we have mentioned, when c = -∇γ, the equation (P λ ) is rewritten as

-div(e γ ∇u) = λe γ f (u). (4.1)
With the variational structure, the stability of minimal solutions u λ is equivalent to

Ω e γ |∇ψ| 2 ≥ λ Ω e γ f (u λ )ψ 2 , ∀ ψ ∈ H 1 0 (Ω). (4.2)
Moreover, for any C 1 functions φ and ξ satisfying φ(0) = ξ(0) = 0 and ξ = φ 2 , the estimate (2.4) is replaced by

Ω e γ f (u λ )φ 2 (u λ ) ≤ Ω e γ f (u λ )ξ(u λ ).
Taking now φ(t) = f (t) -f (0) and working as for Theorem 1 in [START_REF] Ye | On a general family of nonautonomous elliptic and parabolic equations[END_REF], we have

* ∈ H 2 ∩ H 1 0 (Ω). More precisely, Ω f (u λ )f (u λ ) ≤ C, ∀ λ ∈ (0, λ * ]. (4.3)
When Ω = B 1 is the unit ball, γ(x) = γ(r) with r = |x|, u λ is radial by uniqueness of the minimal solution and satisfies

-u - n -1 r u -γ u = λf (u) in (0, 1], (4.4)
with u (0) = 0 and u(1) = 0. Our main result in this section is the regularity of the extremal solution u * for any f satisfying (H) provided n = 2 and the optimal estimate for u claimed in Theorem 1.2.

The method we use is similar to [START_REF] Cabré | Regularity of radial minimizers and extremal solutions of semilinear elliptic equations[END_REF][START_REF] Villegas | Shape estimates for semi-stable radial solutions of semilinear elliptic equations[END_REF], but the uniform boundedness of u λ C 1 is not enough to claim the regularity of u * , because a singular u * could be Lipschitz in many cases (see Remark 1.1). In fact, the estimate (4.3) is crucial for our proof.

As in [START_REF] Cabré | Regularity of radial minimizers and extremal solutions of semilinear elliptic equations[END_REF][START_REF] Villegas | Shape estimates for semi-stable radial solutions of semilinear elliptic equations[END_REF], since u λ (r) ≤ 0 by maximum principle or equation (4.4), the boundedness of u λ H 1 0 implies that for any k ∈ N, r > 0, u λ C k (B1\Br) ≤ C k,r , ∀ λ ∈ (0, λ * ]. So we concentrate our attention near the origin. Derivating the equation (4.4) or (4.1) with respect to r,

-div (e γ ∇u ) = e γ u λf (u) - n -1 r 2 + γ in (0, 1].
Using ψ = rη(r)u λ (r) as test function in (4.2) with η ∈ H 1 0 (B 1 ) ∩ C(B 1 ), by similar calculation as for Lemma 2.1 in [START_REF] Cabré | Regularity of radial minimizers and extremal solutions of semilinear elliptic equations[END_REF], we obtain For simplicity, we drop the index λ. All estimates below hold uniformly for λ. First as u λ is radial, by maximum principle, we see that u is decreasing in r. Since f and f are nondecreasing functions according to (H), the estimate (4.3) implies (as n = 2)

B1 e γ |∇(rη)| 2 -(n -1)η 2 + γ r 2 η 2 u 2 λ ≥ 0, ∀ λ ∈ (0, λ * ]. ( 4
πr 2 f (u(r))f (u(r)) ≤ Br f (u)f (u) ≤ C, ∀ r ∈ (0, 1]. By Lemma 2.1, we have f (u(r)) ≤ C r for all r ∈ (0, 1]. ( 4.6)
Let r 0 ∈ (0, 1 2 ]. Let η be a radial function in

H 1 0 (B 1 ) ∩ C 0 (B 1 ) such that η(r) = r -1 0 if r < r 0 ; r -1 if r 0 ≤ r ≤ 1 2 ,
and η be a fixed C 1 function in B 1 \ B 1/2 , independent of r 0 . The direct calculation yields

|∇(rη)| 2 -η 2 + γ r 2 η 2 = γ r 2 r -2 0 if r < r 0 ; γ -r -2 if r 0 < r ≤ 1 2 .
Using (4.5), as u is uniformly bounded in H 1 (B 1 ) by Proposition 2.4 and r 2 r -2 0 ≤ 1 in [0, r 0 ], we get There exists r 1 > 0 small enough such that Ψ ≥ 0 in [0, r 1 ] with λ ≥ λ * 2 . Using (4.4), (4.6) and (4.9), for λ ≥ λ

Lemma 4 . 1 .

 41 When c = ∇γ, the extremal solution u

. 5 ) 4 . 1 .

 541 Proof of Theorem 1.1

7 ) 2 . 2 ,u

 722 Consider the following test function used in[START_REF] Villegas | Shape estimates for semi-stable radial solutions of semilinear elliptic equations[END_REF]: For any r ≤1 2 and 0 < r 0 < r,η(s) =    (rr 0 ) -1 if s < r 0 ; (rs) -1 if r 0 ≤ s < r; s -2 if r ≤ s ≤ 1Applying again (4.5) and combining with (4.7), we obtain finally (withr 0 → 0) γ ru ) = -λe γ rf (u) with n = 2, so e γ ru is nonincreasing in r. Then u (s) ≤ Cru (r)/s for s ∈ [r, 1], hence u (s) ≤ Cu (r) ≤ 0 for any s ∈ [r, 2r] if r ≤ 1 2 . By (4.8), for any 0 < r ≤ 1 (r) 2 ds = C 3 u (r) 2 .That means|u (r)| ≤ Cr in [0, 1].(4.9)However, we need to consider also u (r) as explained above. LetG(r) = e γ ru and Ψ(r) = -2G( √ r) -M r 0 (r -s)f u( √ s) dswhere M is a constant to be chosen. Using G = -λe γ rf (u),Ψ (r) = λe γ(s) f (u(s)) u (s) 2s + λe γ(s) f (u(s)) γ (s) 2s -M f (u(s)) s= √ r ≤ λe γ(s) f (u(s)) γ (s) 2s -M f (u(s)) s= √ r ≤ C 0 f u( √ r) -M f u( √ r) .For the last line, we used |γ (s)|/s ≤ C in [0, 1] since γ is a smooth function (so γ (0) = 0). Fix M > C 0 + 1, Ψ is then concave in [0, 1]. On the other hand, by (4≥ Cλf (0) -CM √ r.

* 2 and r ≤ r 1 ,u 2

 12 (r)r n-1 dr ≤ Cr 2+2 √ n-1 0 , ∀ r 0 ∈ [0, r 1 ].By the monotonicity of e γ r n-1 u , similarly as for (4.9), it holds|u (r)| ≤ Cr -n 2 +1+ √ n-1 , ∀ r ∈ [0, 1].Finally, combining with (4.10), we are done (in fact, -n 2 + 1 + √ n -1 ≤ -1 for n ≥ 10).

Acknowledgments Part of the work was completed during X.L.'s visit to University of Connecticut (Uconn) with the financial support of CSC. She would like to thank the Department of Mathematics of Uconn for its warm hospitality. She thanks also Prof. Ryzhik for useful discussion. D.Y. is supported by the French ANR project referenced ANR-08-BLAN-0335-01. F.Z. is supported in part by NSFC No. 10971067, the "basic research project of China" No. 2006CB805902 and Shanghai project 09XD1401600.

Applying one more time (4.9), we see that u ( √ r) ≥ -C for any λ ≥ λ * 2 and r ≤ r 1 . Otherwise, by (4.4) and (4.9), u (r) ≤ -u (r)r -1 -γ (r)u (r) ≤ C, we claim then

Combining with (4.4) and (4.9), it means λf (u) ∞ ≤ C, no singularity will occur.

Proof of Theorem 1.2

As above, we drop the index λ and all estimations hold uniformly for λ. First, repeating the proof of Theorem 1.8, c) in [START_REF] Cabré | Regularity of radial minimizers and extremal solutions of semilinear elliptic equations[END_REF], we obtain

Let η be a radial function in

in B r1 and be a fixed C 1 function in B 1 \ B r1 , here r 0 is any constant in (0, r 1 ), r 1 > 0 is a small constant to be determined. Therefore

We fix r 1 > 0 small enough such that max

Tending r 0 to 0, we have

Now we take another test function used in [START_REF] Villegas | Shape estimates for semi-stable radial solutions of semilinear elliptic equations[END_REF],


Q U Óc

Anh Ng Ô

QUOC Van Hoang Nguyen

AND Hung Phan
email: phanquochung@dtu.edu.vn

Dong Ye
email: dong.ye@univ-lorraine.fr

Q H Phan

EXHAUSTIVE EXISTENCE AND NON-EXISTENCE RESULTS FOR SOME PROTOTYPE POLYHARMONIC EQUATIONS IN THE WHOLE SPACE

Keywords: 2010 Mathematics Subject Classification. Primary 35B53, 35J91, 35B33, Secondary 35B08, 35B51, 35A01 Polyharmonic equation, Existence and non-existence, Liouville theorem, Maximum principle type result

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity

with n 1, m 1, and α ∈ R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and α.

Remarkably, we are able to provide necessary and sufficient conditions on the exponent α to guarantee the existence of such solutions in R n . Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere.

INTRODUCTION

In this paper, we are interested in the existence and non-existence results for the following equations ∆ m u = ±u α in the whole Euclidean space R n , (

where n, m 1, and α ∈ R is a parameter. It is easy to see that equations (1.1) can be rewritten in the form (-∆) m u = ±u α .

However, we intend to keep rather the notation ∆ m instead of (-∆) m for the convenience of presentation.

Among others, one basic reason that we are interested in such equations is that (1.1) are the simplest models of polyharmonic equations with power-type nonlinearity. In the literature, equations of the form (1.1) have attracted much attention in various mathematical directions, including the existence and non-existence results, the multiplicity, the regularity, the stability of solutions, the asymptotic behaviors at infinity of entire solutions, as well as Liouville-type results, etc. Since there is a huge number of works related to equations of the form (1.1), it is impossible for us to cover and mention all works and results even if there are some closely related to our work. However, among many references in the literature, we would like to refer the reader to the monograph [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF] for further motivations and results.

The exponent α here can take any value in R. Regarding the nonlinearity u α , it is usually called superlinear, sublinear, or singular respectively if α > 1, α ∈ (0, 1), or α < 0. As we can imagine, the existence of solutions to (1.1) strongly depends on the range of the exponent α, on the dimension n, and on the fact that m is even or odd.

It is well known that the semilinear polyharmonic equations arise in many physics phenomena. For example, several particular cases of (1.1) have their origins such as the elasticity, the equilibrium states for thin films, the modeling of electrostatic actuations, etc.

The equations (1.1) have also their root in conformal geometry. The equation

-∆u = k(x)u n+2 n-2
with n 3 is closely related to the famous Yamabe problem and the prescribing scalar curvature problem. The geometric aspect of higher order cases m 2 are related the problem of prescribing Q-curvature on Riemannian manifolds. Loosely speaking, given a Riemannian manifold (M n , g) of dimension n, we denote by P g m the GJMS operator of order m, constructed by Graham, Jenne, Mason and Sparling in the celebrated work [GJMS92]. The prescribing Q-curvature problem asks us to look for a positive solution u to the following partial differential equation on

M n P g 2m (u) = Q(x)u n+2m n-2m .
Under conformal projection or as the limit equation of blow-up analysis, we are often led to understand the problems like

(-∆) m u = ±u n+2m n-2m
in R n , so a special case of (1.1).

If the second order case m = 1 is well understood, the situation of polyharmonic problems m 2 is much less clear. For example, as far as we know, we cannot find exhaustive results on the existence or non-existence of positive solutions to (1.1) for all exponents α ∈ R. To be clear, by solutions in this paper, we mean the classical solutions. Our main purpose here is to give a complete answer to this question, that is, to establish the existence or the non-existence of positive solutions to (1.1) for any m, n 1, and α ∈ R. We will handle also the case of non-trivial non-negative solutions when α 0, with the natural convention 0 0 = 1.

In other words, we will find the necessary and sufficient conditions on the exponent α to confirm the non-existence, i.e. the Liouville-type results for positive solutions with real exponents α; and the Liouville-type results for non-negative solutions in R n provided α 0. Such results are sometimes called optimal Liouville-type theorems. The reason to consider separately the two classes of solutions is due to the lack of the strong maximum principle for high order elliptic operators, when m 2.

In recent years, the Liouville property has emerged as an important subject in the analysis of nonlinear partial differential equations. In particular, Polacik, Quittner and Souplet [PQS07a, PQS07b] developed a general method to derive universal pointwise estimates of local solutions from Liouville-type results. Their approach is based on rescaling arguments combined with a key doubling property, which is different from the classical rescaling method of Gidas and Spruck [START_REF] Gidas | A priori bounds for positive solutions of nonlinear elliptic equations[END_REF]. It turns out that one can obtain from Liouville-type theorems a variety of results on qualitative properties of solutions, such as a priori estimates, universal bounds, universal singularity and decay estimates, etc. For this reason, we expect to see many applications of Liouville-type theorems obtained here.

Before closing this section, we would like to mention the outline of the paper. The next section is devoted to the statement of our main results, which consist of two theorems. Theorem 2.1 concerns the solvability of (1.1) with a negative sign, that is ∆ m u = -u α , while Theorem 2.2 concerns solutions to ∆ m u = u α . The proofs of Theorems 2.1 and 2.2 are presented in Section 4, where we used several important approaches, including a priori integral estimates derived for local solutions, interpolation inequalities, the comparison principle for radial solutions, the derivation of sub/super polyharmonic properties and the Moser's iteration. In the last section, we identify all the situations where an entire nontrivial, non-negative solution must be positive, by proving Propositions 2.1 and 2.2.

STATEMENT OF MAIN RESULTS

2.1. Some known results. Let us start by reviewing some well-known results concerning the existence and non-existence of solutions to the problems (1.1). We recall the Sobolev exponent

p S (n, m) =    n + 2m n -2m if n 2m + 1, ∞ if n 2m.
(2.1)

The first known result is for positive solutions to (1.1) in the singular case α < 0.

Proposition A. Let m 2 be an integer and n 3. Assume α < -1/(m -1). Then (1.1) always possesses positive solutions.

Proposition A was proved by Kusano, Naito and Swanson via the Schauder-Tychonoff fixed-point theorem. More precisely, as a special case of Theorem 1 in [KNS88], if n 3 and

∞ 0 t(1 + t 2m-2 ) α dt < ∞, (2.2)
then (1.1) possesses infinitely many positive radial solutions satisfying the following growth condition

C 1 (1 + |x| 2m-2 ) u(x) C 2 (1 + |x| 2m-2 ) in R n ,
where C 1 and C 2 are positive constants. It is obvious that the integral in (2.2) is finite when α < -1/(m -1).

Remark 2.1. We stress that the restriction on dimension n 3 in Proposition A is necessary for problem ∆ m u = -u α . This is because the equation does not has any solution when n 2; see the Proposition 4.1 below for the non-existence result.

Next we collect some known results for the equation

(-∆) m u = u α in R n ,
(2.3) in the superlinear case α > 1. These results can be summarized as follows.

Proposition B. Let m be a positive integer. We have the following claims: Let us now comment on Proposition B. Part (i) is commonly known as the subcritical case. From the definition of the Sobolev critical exponent p S (n, m) we know that p S (n, m) is finite if n > 2m. In this setting, the second order case, namely m = 1, was first established by Gidas and Spruck in [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] via the technique of nonlinear integral estimates and the Bochner formula. Chen and Li gave a different proof in [START_REF] Chen | Classification of solutions of some nonlinear elliptic equations[END_REF] by using the moving plane method combined with the Kelvin transform.

(i) If 1 < α < p S (
For higher order cases, Lin resolved in [Lin98] the case m = 2 and it was finally generalized by Wei and Xu in [WX99] for any m 2 via the argument of moving planes. For the remaining case n 2m and with arbitrary m, the non-existence result (i) can be deduced from the method of rescaled test-function [MP01]; and also from the method of representation formula as presented in [CDM08]. Now we turn to the Part (ii). The case m = 1 can be proved easily by applying the Pohozaev identity [Poh65] with radial solutions on balls, or it can be deduced from the shooting argument [JL73]. In addition, if α = p S (n, 1), the critical exponent, it was showed by Caffarelli, Gidas and Spruck in [CGS89] that any positive solution to

-∆u = u n+2 n-2 in R n
with n 3 is radially symmetric up to translation, and it is unique up to dilation.

The above classification was extended to the case of biharmonic equation by Lin [Lin98] and to the general case m 2 by Wei and Xu [WX99]. More precisely, it was shown in [WX99] that any positive solution u to

(-∆) m u = u n+2m n-2m in R n with n > 2m 2 is of the following form u(x) = 2λ 1 + λ 2 |x -x 0 | 2 n-2m 2
for some x 0 ∈ R n , λ > 0.

Let us now turn to the supercritical case, namely α > p S (n, m). When m = 1, the existence of positive solutions to (2.3) was obtained by Ni in [Ni82, Theorem 4.5]. For the higher order case, namely m 2, the existence of positive solutions to (2.3) was shown by Liu, Guo and Zhang in [LGZ06, Theorem 1.1]. They used a combination of the shooting method together with degree theory and the Pohozaev identity.

However, it becomes evident from the detailed description mentioned above that after putting together all the known results in the literature, the knowledge on this class of equations still appears quite fragmentary. Our aim in this paper is to consider all the situations m 2, n 1 and α ∈ R, and determine whether positive or non-negative solutions of (1.1) exist.

For the sake of transparent presentation, we shall present our results in two different subcases according to the sign of the right-hand side. We will also see that the range of α insuring the existence of solutions to equations (1.1) strongly depends on the parity of m.

2.2.

Exhaustive results for ∆ m u = -u α in R n . As mentioned above, the results depend on the parity of m, it is more convenient to split the study for two equations:

∆ 2k u = -u α in R n (P - 2k ) and ∆ 2k-1 u = -u α in R n , (P - 2k-1 )
where k is a positive integer. For (P - 2k ), as far as we know, there are many results which are limited to the case k = 1, i.e. for the biharmonic equation

∆ 2 u = -u α in R n .
(2.4)

Here, the non-existence of positive solutions to (2.4) with α ∈ [-1, 0] was first proved by Choi and Xu in [CX09, Theorem 1.1] for dimension n = 3. This result was extended to all dimensions by Lai and Ye in [LY16, Theorem 1.3]. On the other hand, Proposition A, see also [MR03, Theorem 3.1], ensures the existence of positive solutions for any α < -1.

For the problem (P - 2k-1 ), when k = 1, it is worth noticing that the class of positive solutions coincides with the one of non-trivial non-negative solutions, due to the strong maximum principle. For k = 1, the non-existence of positive solution when α 1 is well known, see for instance the results in [DM10, Theorem 2.7] and [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]; while the situations α > 1 was fully settled in [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF]. Recently, it was proved in [DN17] that

∆ 3 u = -u α in R 3 has no positive solution if α ∈ [-1/2, 0).
We give here a complete answer to the question of existence for ∆ m u = -u α . For convenience, we use the convention that -1/(m -1) = -∞ when m = 1. Our first result reads as follows:

Theorem 2.1. Let m be a positive integer. Then we have the following claims:

(i) The problem ∆ m u = -u α possesses a positive solution if and only if either n 3 and α < -1/(m -1); or m is odd and α p S (n, m). (ii) The problem ∆ m u = -u α with α 0 has a non-trivial non-negative solution if and only if m is odd and α p S (n, m).

The existence of positive or non-trivial non-negative solution to ∆ m u = -u α can be easily summarized in the following table.

α < -1 m-1 -1 m-1 α < 0 0 α 1 α > 1 u > 0 n 2 NO NO
∆ m u = -u α in R n
Recall that we are concerned with classical solutions, then for α < 0, there is no proper non-negative solution, out of positive solutions. That's the reason of the above grey cells.

As the equation (P -

2k-1 ) with α p S (n, m) always admits positive solutions, hence non-trivial, non-negative solution. A natural question for (P - 2k-1 ) is that whether or not there is non-trivial, non-negative but not positive solution, the following maximum type result indicates that such solution does not exist.

Proposition 2.1. Let m 1 and α > 1. Then any non-trivial, non-negative solution to the equation ∆ m u = -u α in R n must be positive everywhere.

Clearly, our contributions in Theorem 2.1 are multifold:

• By determining the sign of ∆ m-1 u with Lemma 3.3, we show quickly the nonexistence of positive solution to The proof of the non-existence in the singular case α ∈ [-1/(m -1), 0) with n 3 relies on the convexity of the function t → t α and comparison principle. In the superlinear case α > 1 and for the equation (P - 2k ), we made use of the integral estimate and a Liouville type result; see Lemma 3.4. However, the case α ∈ [0, 1] is significantly more delicate despite the fact that a nonexistence result in the case m = 1 is already known. For example, among others, Mitidieri and Pohozaev used the standard rescaled test-function method to obtain the non-existence result in [MP01, section 12].

∆ m u = -u α in R n with n = 1,
For arbitrary m, it seems that many well-known approaches, such as the classic rescaled test-function method [MP01], the moving plane technique [START_REF] Chen | Classification of solutions of some nonlinear elliptic equations[END_REF], the argument of maximum principle [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF], the representation formula method [CDM08], or the derivation technique of super/sub polyharmonic property [Lin98, WX99, CL13, Ngo17], cannot be applicable.

Our proof in the sublinear case is inspired by the idea of Serrin and Zou [SZ96] and Souplet [Sou09], it is based on the integral estimates and Moser's iteration method; see the proof of Propositions 4.3.

Exhaustive results for

∆ m u = u α in R n .
We give here a complete answer to the question of existence for (1.1) with the plus sign. Our second result reads as follows:

Theorem 2.2. Let m be a positive integer. Then we have the following claims. As before, we will split our study into two equations according to the parity of m, that is,

∆ 2k u = u α in R n (P + 2k
) and

∆ 2k-1 u = u α in R n , (P + 2k-1 )
where k is a positive integer. Our contributions are twofold here:

• We give a unified proof of the existence of positive solutions for all α 1; see Proposition 4.5.

• We prove the non-existence of non-trivial, non-negative solutions for (P + 2k-1 ) with any α > 1; see Proposition 4.4.

In the second order case, it is well known that the problem ∆u = u α in R n has no positive solution if α > 1, but it possesses a positive one if α 1. More precisely, the non-existence for the superlinear case α > 1 is a consequence of the so-called Keller-Osserman criteria developed by Keller [START_REF] Joseph | On solutions of ∆u = f (u)[END_REF] and Osserman [Oss57]. Their theory can be employed to show that the equation ∆u = u α admits no non-trivial, non-negative, entire solution whenever α > 1, see also [START_REF] Brezis | Semilinear equations in R N without condition at infinity[END_REF]Lemma 2].

When α 1, the existence of radial solutions can be easily obtained by the monotonicity of u(r). We note that for m 2, we can also apply Proposition A to obtain the existence of solutions to (P + 2k ) and (P - 2k ) for α < -1/(m -1).

Similarly to the question raised for (P - 2k-1 ), we can ask here if the set of non-trivial, non-negative solutions and the set of positive solutions coincide. We provide a complete answer as follows.

Proposition 2.2. Let m be a positive integer and α 0. Then the equation ∆ m u = u α possesses entire, non-trivial, non-negative but not strictly positive solution in R n if and only if α ∈ [0, 1] and (α, m) = (1, 1).

In other words, if either α > 1 or (α, m) = (1, 1), then any entire, non-trivial, nonnegative solution to ∆ m u = u α must be positive everywhere.

Before closing this section, we would like to comment on Propositions 2.1 and 2.2. From our point of view, these results can be regarded as maximum principle results. As far as we know, similar results do exist in the literature, however, with some limitations, see for example [CDM08].

PRELIMINARIES

In what follows, the notation ∆ i u stands for u when i = 0. The notation B r is always understood as the open ball B r (0) centered at the origin with radius r. When u is a radial function, instead of writing u(x), we also use the notation u(r). Throughout the paper, we also use the notation u(r) to denote the spherical average of u centered at the origin on the sphere ∂B r , the boundary of the ball B r , that is

u(r) = 1 |∂B r | ∂Br udσ.
Spherical averaging has some nice properties such as it enjoys

u(r) = 1 |∂B 1 | ∂B1 u(rω)dσ ω
and the following identity ∆u = ∆u for any C 2 -function u. This simply follows from ∆u = r 1-n (r n-1 u ) and the following

u (r) = 1 |∂B 1 | ∂B1 ω • ∇u(rω)dσ ω = 1 |∂B 1 |r n-1 Br ∆udx = 1 r n-1 r 0 s n-1 ∆u(s)ds.
Throughout the paper, the symbol C denotes a generic positive constant whose value could be different from one line to another.

Here are some basic results, which will be useful for our analysis.

Lemma 3.1. Let m 1 and v 1 , v 2 : B R → I ⊂ R be two C 2m radial functions verifying ∆ m v 1 f (v 1 ), ∆ m v 2 f (v 2 ) in B R and ∆ i v 1 (0) ∆ i v 2 (0), ∀ 0 i m -1. If f is non-decreasing in I, then v 1 v 2 in B R . In other words, v 1 (r) v 2 (r) for all r ∈ [0, R).
The above comparison principle is a special form of more general well-known results; see for instance [FF16, Proposition A.2] or [LS09, Remark 2.3]. An easy consequence of the above comparison principle is the following pointwise estimate.

Lemma 3.2. Let u be in C 2m (R n ) satisfying ∆ m u 0 in R n , then we have u(r) u(0) + m-1 i=1 ∆ i u(0)r 2i 1 k i [2k(n + 2k -2)]
, ∀ r 0.

(3.1)

Proof. Let Φ be the radial function defined by

Φ(r) := u(0) + m-1 i=1 ∆ i u(0)r 2i 1 k i [2k(n + 2k -2)]
.

Using ∆u = ∆u, we deduce by induction that ∆ m u = ∆ m u. There hold then

∆ m Φ ≡ 0 ∆ m u = ∆ m u in R n and ∆ i Φ(0) = ∆ i u(0) = ∆ i u(0) for any 0 i m -1.
Applying Lemma 3.1 with f ≡ 0, there holds u Φ in R n .

By an elementary computation involving the Gamma function, it is easy to verify that

1 k p [2k(n + 2k -2)] = 2 2p p!Γ p + n 2 /Γ n 2 , ∀ p, n ∈ N * .
Therefore, the right hand side of (3.1) is nothing but the main part of classical Pizzetti's expansion formula in [Piz09]; see also Equation (8) in Nicolesco's paper [Nic32]. In Pizzetti's formula, there is a last term involving ∆ m u. We can remark that if ∆ m u 0 in R n , then the remained term in Pizzetti's formula is non-positive, which implies then (3.1). Nevertheless, our proof of (3.1) is simple and constructive.

The following result is a simple but important fact of our approach.

Lemma 3.3. Let m 1. Then we have the following claims:

(i) If u be a positive function satisfying ∆ m u < 0 in R n , then ∆ m-1 u > 0 in R n . (ii) If u be a non-negative function satisfying ∆ m u 0 in R n , then ∆ m-1 u 0 in R n .
Proof. Consider first the claim (ii). Set w = ∆ m-1 u, suppose that there is some x 0 ∈ R n such that w(x 0 ) < 0. By a translation, we may assume that x 0 = 0. Moreover, it follows from Lemma 3.2 that u satisfies the estimate (3.1). As ∆ m-1 u(0) < 0, there holds u(r) < 0 for r large enough. This is impossible because u is non-negative in R n . The point (ii) holds true. Now we consider (i). Set again w = ∆ m-1 u, we have w 0 in R n by (ii). If w vanishes at some x 1 ∈ R n , then w attains its minimum at x 1 . However, this contradicts the fact that ∆w(x 1 ) < 0, we are done.

It is worth noting that without the non-negativity of u, in general, the result of Lemma 3.3 does not hold. For example, it was shown in [FF16, Lemma 7.8] that there are infinitely many entire radial solutions to ∆ 2k+1 u = -e u for which ∆ 2k u changes sign.

The following Liouville type result is a crucial step in the proof of Proposition 4.4.

Lemma 3.4. Assume that u is a C 2m non-negative function in R n , verifying (-∆) m u 0 in R n and B R udx = o(R n ) as R → ∞.
(3.2)

Then u ≡ 0 in R n . Proof. Let v k = (-∆) k u, for 0 k m.
We shall prove by backward induction on k that

v k 0 in R n (3.3)
for k = m, m -1, . . . , 0. It is obvious that (3.3) is true for k = m. Suppose now (3.3) is true for j + 1 k m with some j 0, we shall show that v j 0 in R n . Depending on the parity of j, we have two possible cases.

Case 1: j is odd. As ∆ j+1 u = v j+1 0, Lemma 3.3 (ii) gives v j 0.

Case 2: j is even. We will prove v j 0. By way of contradiction, assume that there exists some x 0 ∈ R n such that v j (x 0 ) > 0. Up to a translation, we may further assume that x 0 = 0. Then ∆v j = ∆v j = -v j+1 0 in R n . From this we have v j (r) 0 for any r 0, hence

v j (r) v j (0) = v j (0) > 0.
Let ψ be a smooth, radial, cut-off function satisfying 0 ψ 1 and

ψ(x) = 0 if |x| 2, 1 if |x| 1. (3.4)
On one hand, for any R > 0, we can estimate

R n v j (x)ψ x R dx B R v j (x)dx = C R 0 v j (r)r n-1 dr CR n v j (0). (3.5)
On the other hand, there holds

R n v j (x)ψ x R dx = R n (-∆) j u(x)ψ x R dx = R -2j R n u(x)(-∆) j ψ x R dx CR -2j B 2R u(x)dx.
(3.6) Putting (3.5) and (3.6) together gives

0 < v j (0) CR -2j-n B 2R u(x)dx.
Now letting R → ∞ and using (3.2) we meet a contradiction. Hence, we get v j 0 in R n . Therefore, by the induction principle, (3.3) is true as claimed. Taking k = 0 in (3.3), we have u 0 in R n , hence u ≡ 0 in R n .

The last result in this subsection is a classical interpolation-type estimate, which plays an important role in our proof of the non-existence result for ∆ m u = -u α with 0 < α < 1; see the proof of Proposition 4.3. Lemma 3.5. Let m be a positive integer. Let z be a function in W 2m, (B 2R ) for some > 1. Then for any exponent 1 < q < ∞ such that

1 q 1 - 2m n , (3.7)
there holds

B R z q dx 1/q CR n q +2m-n B 2R |∆ m z| dx 1/ + CR n q -n B 2R zdx,
where C = C(m, n, , q).

Proof. By the dilation w(x) = z(Rx), we obtain

B R z q dx = R n B1 w q dx, B 2R zdx = R n B2 wdx,
and

B 2R |∆ m z| dx = R -2m +n B2 |∆ m w| dx.
From these identities, the desired inequality is equivalent to

w L q (B1) C ∆ m w L (B2) + C w L 1 (B2)
for w ∈ W 2m, (B 2 ). However, this follows from (3.7) and standard elliptic estimate; see for instance [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Theorem 2.20]. The lemma is proved.

PROOF OF THE MAIN RESULTS

This section is devoted to the proof of our main results. We prove some Liouville type results in subsections 4.1 and 4.2, while some existence results are proved in subsection 4.3. It is worth noticing that for each case in Tables 1 and 2 above, we have already included the name of the main proposition yielding the result in the case. Therefore, there is no need to write a proof for Theorems 2.1 and 2.2. 4.1. Non-existence results for ∆ m u = -u α . This subsection is devoted to the nonexistence results in Theorem 2.1, and we do not consider specially the situations under applications of Propositions A and B. 4.1.1. For dimensions 1 and 2. Let us start with the case n 2 and this corresponds to the second and fifth rows in Table 1. We will prove that in dimensions one and two, the equation

∆ m u = -u α in R n (4.1)
has no positive solution for any α ∈ R; and has no non-trivial, non-negative solution for any α 0. In fact, these claims are trivial consequences of the following result.

Proposition 4.1. Let m be a positive integer and n 2.

If u is a non-negative C 2m - function verifying ∆ m u 0 in R n , then ∆ m u ≡ 0 in R n . Proof. As ∆ m u 0 in R n , Lemma 3.3 (ii) shows that ∆ m-1 u =: w 0.
This means that w is a non-negative, super-harmonic function in R n . It is well known that such w must be constant in R n if n 2; see [Far07, Theorem 3.1] for the case n = 2. Hence, we must have ∆ m u = ∆w = 0 everywhere.

For negative values of α.

Here we prove the non-existence of positive solution to (4.1) for n 3 and suitable negative values of α. Proof. Assume that n 3, m 1, and α ∈ [-1/(m-1), 0]∩R. By way of contradiction, suppose that u is a positive solution to (4.1). Using Lemma 3.2, we have

u(r) u(0) + m-1 i=1 ∆ i u(0)r 2l 1 k i [2k(n + 2k -2)] ∀ r 0.
Hence, there exists a constant C > 0 such that u(r) Cr 2(m-1) for any r 1.

Set w = ∆ m-1 u. By Lemma 3.3 (i), there holds w > 0. Moreover, as the map t → t α is convex in (0, ∞) when α 0, Jensen's inequality implies

-∆w = u α u α in R n , so that -r n-1 w (r)
r n-1 u α (r) Cr n-1+2(m-1)α , ∀ r 1.

Integrating over (1, r), taking into account n 3 and (m -1)α -1, there holds w (1) -r n-1 w (r) Cr 2(m-1)α+n -C.

Therefore, w (r) -Cr 2(m-1)α+1 + Cr -n+1 , ∀ r 1. (4.2) Depending on the size of α, there are two cases:

Case 1: If α ∈ (-1/(m -1), 0), then integrating (4.2) over [1, r] gives w(r) -w(1) -Cr 2(m-1)α+2 + C, ∀ r 1.
We have then w(r) → -∞ as r → ∞, which is a contradiction with w > 0.

Case 2:

If α = -1/(m -1), then integrating of (4.2) over [1, r] gives w(r) -w(1) -C r 1 r -1 dr + r 1 Cr -n+1 dr = -C ln r + C,
which also implies that w(r) → -∞ as r → ∞. We reach again a contradiction. 4.1.3. For 0 α 1. Now we turn to the case of non-negative, sublinear α. The following non-existence result is one of the main contributions of this paper. Proof. In view of Proposition 4.1, it suffices to consider the case n 3. Depending on the size of α, we consider two possible cases. When α = 0, the equation (4.1) becomes ∆ m u ≡ -1, therefore the non-existence of entire, non-negative solution in R n is a direct consequence of Lemma 3.2, since there exists C > 0 such that u + Cr 2m is polyharmonic, whose average grows at most as r 2m-2 .

From now on, we only consider α ∈ (0, 1]. For convenience, we divide the proof into three steps.

Step 1. Suppose that u is a non-trivial, non-negative solution to ∆ m u = -u α in R n . By Lemma 3.2, we have

u(R) CR 2(m-1) for R 1. Hence B 2R udx =: F (R) CR n+2(m-1) for any R 1. (4.3)
Here C is a constant independent of R. Note that to get the estimate (4.3) we only use the sign of ∆ m u. Now via the rescaled test-function argument we fully use the equation ∆ m u = -u α to estimate F (R) from below; see (4.5). Let ψ be a smooth cut-off function satisfying 0 ψ 1 and (3.4). For any R > 0, let

φ R (x) = ψ 2m+1 x R .
It is not hard to verify the pointwise estimate

∆ m (ψ 2m+1 ) Cψ
for some constant C > 0. Therefore, we can estimate

|∆ m φ R (x)| = R -2m ∆ m (ψ 2m+1 ) x R CR -2m ψ x R = CR -2m φ 1/(2m+1) R (x). (4.4) Hence R n u α φ R dx = - R n ∆ m u φ R dx = - R n u ∆ m φ R dx CR -2m R n u φ 1/(2m+1) R dx.

This yields

B R u α dx CR -2m F (R), ∀ R > 0.
(4.5)

Now we further examine F . In view of (4.3), the function F has at most algebraic growth at infinity. Therefore, it must be doubling along a sequence R i → ∞. We turn this observation into a claim as follows

∃ M > 0 and R i → ∞ such that F (2R i ) M F (R i ) ∀ i. (4.6)
Indeed, assume that (4.6) is false. Let us fix M 0 > 2 n+2m-2 , then there exists R 0 > 0 such that

F (2R) M 0 F (R), ∀ R R 0 . (4.7)
Let R 1 1 be sufficiently large verifying F (R 1 ) > 0, such a R 1 exists since u is nontrivial. Denote R * := max{R 1 , R 0 }. Iterating (4.7) and thanks to (4.3), we arrive at, for any i,

M i 0 F (R * ) F (2 i R * ) C(2 i R * ) n+2m-2 , that is M 0 2 n+2m-2 i CR n+2m-2 * F (R * ) ∀ i.
But this is just impossible if i is large enough by the choice of M 0 . So the claim (4.6) holds true. We are now ready to prove the result for α ∈ (0, 1]. The two cases α = 1 and 0 < α < 1 must be considered separately.

Step 2. Consider first the case α = 1. It follows from (4.5) with R = 2R i and (4.6) that

F (R i ) CR -2m i F (2R i ) CM R -2m i F (R i ), ∀ i. Therefore F (R i ) = 0 for i large enough because R i → ∞, which is a contradiction since u is non-trivial.
Step 3. Here we handle the case α ∈ (0, 1). The idea of our proof in this case is as follows. First, the integral F (R) can be estimated by interpolating between u α L 1 (B 2R )

and u q L 1 (B 2R ) with q > 1 large; see (4.11). Second, u α L 1 (B 2R ) is estimated by (4.5) in terms of R -2m F (R). Moreover, we will estimate u q L 1 (B 2R ) by suitable powers of R, through a Moser type iteration procedure starting with (4.3) for q = 1. This will eventually lead, along a doubling sequence R = R i mentioned in (4.6), to a control of F (R) by a power of R which turns out to be negative; see (4.12). Hence, there holds u ≡ 0 by sending R i → ∞.

We now proceed with the details. Let (q h ) be the sequence defined as follows

q 0 = 1, 1 q h = α q h-1 - 2m n , h = 1, 2, ...
By induction, we can compute q h explicitly as

1 q h = α h - 2m(1 -α h ) n(1 -α)
whenever q h is well defined.

Obviously, the sequence (q -1 h ) is decreasing since α ∈ (0, 1), and there exists a unique integer j * 0 such that 1 q j * +1 0 < q j * .

We will estimate u L q h (B R ) successively. First, for all 0 h j * and R 1, we claim that

B R u q h dx 1/q h CR (n+2m-2)α h . (4.8)
The inequality (4.8) for h = 0 follows from (4.3). Assume that (4.8) is true up to h -1 with h j * . Using the equation ∆ m u = -u α , Lemma 3.5, and (4.3), we get that

B R u q h dx 1/q h CR n q h +2m-nα q h-1 B 2R |∆ m u| q h-1 /α dx α/q h-1 + CR n/q h -n B 2R udx CR n q h +2m-n( 1 q h + 2m n ) B 2R u q h-1 dx α/q h-1 + CR n/q h +2(m-1) = C B 2R u q h-1 dx α/q h-1
+ CR n/q h +2(m-1) .

(4.9)

Thanks to the induction hypothesis on u L q h-1 (B R ) , for any R 1, it follows from (4.9) that

B R u q h dx 1/q h C B 2R u q h-1 dx α q h-1 + CR n/q h +2(m-1) = CR (n+2m-2)α h + CR n/q h +2(m-1)
CR (n+2m-2)α h .

For the last line, we have used

(n + 2m -2)α h - n q h + 2m -2 = (1 -α h ) 2m 1 -α -2(m -1) > 0
to absorb the term R n/q h +2(m-1) into the term R (n+2m-2)α h . Hence the claim (4.8) holds true for all h j * and R 1. Furthermore, by the definition of j * , there holds α q j * -2m n 0.

Proof. Assume that u is a non-negative solution in R n of (4.13) with α > 1. We first derive an integral estimate of u over B R . Let ψ be a smooth, radial, cut-off function satisfying 0 ψ 1 and (3.4). For any R > 0, let φ R (x) = ψ p (R -1 x)

with p = 2mα α-1 > 2m. As for (4.4), the pointwise estimate

|∆ m (ψ p )| Cψ p-2m
holds for some constant C > 0. Hence, we eventually have

|∆ m φ R (x)| CR -2m ψ p-2m x R = CR -2m φ 1/α R (x)
. Therefore, similar to the estimate after (4.4), we obtain the following

R n u α φ R dx = - R n u(-∆) m φ R dx CR -2m R n uφ 1/α R dx.
(4.14) Using Hölder's inequality and noting the support of φ R ,

R n uφ 1/α R dx CR n(α-1) α R n u α φ R dx 1/α . (4.15)
Putting together (4.14) and (4.15), we arrive at

B R udx R n uφ 1/α R dx CR n-2m α-1 .
Now applying Lemma 3.4, we deduce that u ≡ 0 in R n .

4.3.

Existence results for ∆ m u = u α . This subsection is devoted to the existence results for the equation

∆ m u = u α in R n (4.16)
under the condition α 1, in other words, we do not consider the situations under applications of Propositions A and B.

Proposition 4.5. For any positive integers m, n and α 1, the equation (4.16) has infinitely many positive, radial solutions.

Proof. We look for radial solutions of (4.16). To this purpose, consider the following initial value problem

     ∆ m u(r) = u α (r) in [0, R), u(0) = 1; ∆ i u(0) = a i > 0, 1 i m -1, (∆ i u) (0) = 0, 0 i m -1.
(4.17)

Clearly, using standard ODE theory, (4.17) has a unique positive solution in a maximal interval [0, R max ). To get an entire solution to (4.16), we need only to prove that R max = ∞. To do this, we will construct suitable entire sub-and super-solutions to (4.17), then apply the comparison principle. Indeed, let u * (r) ≡ 1.

Trivially, u * is a sub-solution to (4.17) and ∆ i u * (0) ∆ i u(0) for any 0

i m -1. Hence u(r) 1 = u * (r) in [0, R max )
by Lemma 3.1 with f ≡ 0, which is clearly non-decreasing. Now we turn our attention to the existence of a super-solution u * (r). Let

v(r) = e r 2 /2 . It is not hard to see that ∆ k v(r) = P k (r)e r 2 /2
for some function P k . A direct computation yields that P k+1 (r) = (r 2 + n)P k + 2rP k (r) + ∆P k (r) ∀ k 0, r 0.

As P 0 ≡ 1, by induction, we can readily prove that each P k is a polynomial whose coefficients are natural numbers and deg(P k ) = 2k. Because

P k+1 (0) = nP k (0) + ∆P k (0) = nP k (0) + nP k (0)
we can easily see that P k (0) 1 for all k. Now we let u * (r) = λv(r)

with λ = max(1, a 1 , . . . a m-1 ) 1. It follows that ∆ m u * (r) = λP m (r)e r 2 /2 λP m (0)e r 2 /2 λe r 2 /2 , ∀ r 0 (4.18) and ∆ i u * (0) = λP i (0) λ ∆ i u(0) for 0 i m -1.
There are two possibilities:

Case 1: α ∈ [0, 1]. In this case, (4.18) yields ∆ m u * (r) λ α e αr 2 /2 = u * (r) α for r 0.

Therefore, u * is indeed a super-solution to (4.17). Then we can apply Lemma 3.1 with

f (t) = t α in R + to get that u(r) u * (r) in [0, R max ).
Case 2: α < 0. In this case, on one hand, (4.18) yields ∆ m u * 1. On the other hand, we have already shown that u 1 in [0, R max ), which immediately yields 1 u α = ∆ m u in B Rmax . Therefore, this time applying Lemma 3.1 with f ≡ 1, now there holds u(r) u * (r) in [0, R max ).

Combining two cases, whenever α 1 we can always conclude that u * u u * in B Rmax , where u * and u * are smooth positive functions in R n constructed above. As u is locally uniformly bounded, we can obtain the local boundedness of all derivatives of u up to order 2m -1 by successive integrations; see [START_REF] Farina | Existence and stability properties of entire solutions to the polyharmonic equation (-∆) m u = e u for any m ≥ 1[END_REF]Proposition A.2]. This readily implies that R max = ∞ as claimed.

The infinity of solutions can be obtained by choosing different values of a i if m 2; or at least by the natural scaling of the equation (4.16).

Remark 4.1. If n 2, then we can also put solutions of lower dimensions in R n to get infinitely many non radial solutions to (4.16) with α 1. Similar remark goes to the equation (4.1) when n 4.

MAXIMUM PRINCIPLE TYPE RESULTS

This section is devoted to proofs of Propositions 2.1 and 2.2. We use an elementary property for non-negative super-polynharmonic radial functions.

Lemma 5.1. Let m 1. Assume that w is non-trivial, non-negative, radial function such that (-∆) m w 0 in R n . Then, either w(0) > 0 or lim r→∞ w(r) = ∞.

Proof. Let m = 1, if w is non-negative, non-trivial, and -∆w 0, then the strong maximum principle yields w(0) > 0. Suppose that the conclusion is true up to some positive integer m, we consider now w such that (-∆) m+1 w 0 in R n .

We have two cases:

Case 1: If m + 1 is odd, then by using Lemma 3.3 (ii), there holds ∆ m w 0, namely (-∆) m w 0 in R n . From this we get the result by induction hypothesis.

Case 2: If m + 1 is even, then we have ∆ m+1 w 0, which means that ∆ m w(r) is non-decreasing in r. Therefore, there exists lim r→∞ ∆ m w(r) = ∈ R ∪{∞}.

If > 0, then we readily have lim r→∞ w(r) = ∞ by comparison principle. If 0, then there holds ∆ m w 0 everywhere, namely (-∆) m w 0 in R n . Again we conclude with the induction hypothesis.

Combing the above two cases, we are done.

5.1.

A maximum principle type result for ∆ m u = -u α . We are now in a position to prove Proposition 2.1. Let u be a non-trivial, non-negative solution to ∆ m u = -u α . In view of Theorem 2.1 we are limited to the case where m is odd with α p S (n, m) > 1.

By way of contradiction, suppose that u(x 0 ) = 0 for x 0 ∈ R n , without loss of generality, we can assume that x 0 = 0. Following the proof of Proposition 4.4, there holds B R udx CR n-2m α-1 .

(5.1)

Taking the average over spheres, we get u(0) = 0, and more importantly

∆ m u = -u α 0 in R n .
Since u is non-trivial and non-negative, so is u. As u(0) = 0 and m is odd, applying Lemma 5.1, we get lim r→∞ u(r) = ∞, which contradicts with (5.1) if R → ∞.

5.2.

A maximum principle type result for ∆ m u = u α . Now we consider ∆ m u = u α with α 0. We prove Proposition 2.2. In view of Theorem 2.2, we are limited to the case either 0 α 1; or m is even and α p S (n, m).

Firstly, for the case 0 α 1 and m 2, we can construct easily non-trivial nonnegative radial solutions with u(0) = 0. Indeed, consider the similar initial value problem to (4.17) where we choose u(0) = 0, ∆u(0) = a 1 > 0, ∆ i u(0) = a i 0, i = 2, . . . m -1.

Then we get a global solution which is non-trivial because it verifies u(r) u * (r) := a 1 2n r 2 in R n .

Another easy fact is that when α ∈ [0, 1) and m 1. As 2m 1-α 2m, there exists C > 0 depending on m, n, and α such that Cr 2m 1-α is an entire classical solution for ∆ m u = u α in R n . Remark that the function r 2m 1-α belongs to C 2m (R n ).

Consider now the case α > 1 and m is even. The proof, based on a contradiction argument, is very similar to the ∆ m u = -u α case with m odd. Let u be a non-trivial nonnegative solution verifying u(0) = 0, then the estimate (5.1) remains valid. As (-∆) m u 0 in R n and u(0) = 0, by Lemma 5.1, lim r→∞ u(r) = ∞, which is impossible seeing (5.1).

It remains to consider m = α = 1. Suppose that a classical non-negative solution to ∆u = u exists in R n , and u(x 0 ) = 0. We can assume that x 0 = 0. Clearly, u is still a classical solution to the same equation, hence u(r) is nondecreasing. By direct integration u (r) = r 1-n r 0 s n-1 u(s)ds r n u(r), ∀ r 0.

From this, the Gronwall inequality yields u(r) u(0)e r 2 2n , hence u 0 as u(0) = 0. We immediately have u ≡ 0 in R n , so is u, which is absurd. This completes our proof.

  n, m) then the problem (-∆) m u = u α has no non-trivial nonnegative solution in R n . (ii) If n > 2m and α p S (n, m), then the problem (-∆) m u = u α possesses positive radial solutions in R n .

  2 for any α ∈ R; see Proposition 4.1. • For any m 1 and n 3, we obtain the non-existence of positive solution in the range α ∈ [-1/(m -1), 1]; see Propositions 4.2 and 4.3. • We obtain the non-existence of non-trivial, non-negative solution in the range α ∈ [0, ∞) for the equation (P - 2k ); see Propositions 4.3 and 4.4.

  (i) The problem ∆ m u = u α possesses a positive solution if and only if either α 1 or m is even and α p S (n, m). (ii) The problem ∆ m u = u α with α 0 has a non-trivial non-negative solution if and only if either 0 α 1 or m is even and α p S (n, m).The results of Theorem 2.2 are summarized in the following table.

  iff m is even and α p S (n, m) Props. B and 4.4 Prop. 4.5 u 0 TABLE 2. Existence results for the problem ∆ m u = u α in R n

  Proposition 4.2. Let n 3. Then the equation (4.1) has no positive solution for any α ∈ [-1/(m -1), 0) if m > 1; and for any α < 0 if m = 1.

Proposition 4. 3 .

 3 For any n 1, m 1, and α ∈ [0, 1], the equation (4.1) has no non-trivial, non-negative solution.

TABLE 1 .

 1 Existence results for problem

	NO	NO

ACKNOWLEDGMENTS QAN would like to thank Professor Yoshikazu Giga and Professor Kazuhiro Ishige for their useful comments and suggestions on this research project. This project was initiated while QAN was visiting the Vietnam Institute for Advanced Study in Mathematics (VI-ASM) in 2017. He would like to thank the institute for the hospitality and support during the visit. The research of QAN was partially funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.02 and is supported by the Tosio Kato Fellowship awarded in 2018. The research of VHN was partially funded by the Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology. The research of QHP is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.307. DY is partially supported by Science and Technology Commission of Shanghai Municipality (STCSM) under grant No. 18dz2271000. Last but not least, the authors would like to thank an anonymous referee for careful reading of the paper and valuable comments, which helps us to improve the paper. ORCID IDS Qu óc Anh Ngô: 0000-0002-3550-9689, Van Hoang Nguyen: 0000-0002-0030-5811

Therefore, for any q > 1, the condition (3.7) is always fulfilled with = q j * /α. Applying Lemma 3.5, (4.8) with h = j * , the expression of q j * and (4.3), we get

and apply Hölder's inequality with help from (4.5) to obtain

(4.11)

Now we apply (4.11) for the sequence (R i ) satisfying the doubling property (4.6) to get

Combining the above estimate with (4.10), we get

(4.12)

We fix q > 1 large enough such that

Then the estimate (4.12) implies that F (R i ) → 0 as i → ∞. Immediately, this is a contradiction because u is non-trivial.

4.2. Non-existence results for (-∆) m u = -u α with α > 1. In this subsection, we consider non-negative classical solutions of

under the restriction α > 1. This corresponds to the last column of Tables 1 and2.

Proposition 4.4. For any n 1, m 1, and α > 1, the equation (4.13) has no non-trivial, non-negative solution.


Juncheng Wei
email: wei@math.cuhk.edu.hk

Dong Ye
email: dong.ye@univ-metz.fr

On MEMS equation with fringing field

Keywords: MEMS, rupture, fringing field, bifurcation 2000 Mathematics Subject Classification. 35B45, 35J40

We consider the MEMS equation with fringing field

.

Introduction

We consider the following elliptic equation

(E λ ) -∆u = λ(1 + δ|∇u| 2 ) (1 -u) 2
in Ω, u = 0 on ∂Ω, where δ, λ are positive constants, and Ω is a bounded smooth domain in R n (n ≥ 2).

Problem (E λ ) arises in the study of electrostatic Micro-Electromechanical System (MEMS) device. We refer to [START_REF] Flores | Dynamics and touchdown in electrostatic MEMS[END_REF] and the book [START_REF] Pelesko | Modeling MEMS and NEMS[END_REF] for detailed discussions on MEMS devices modeling. The parameter λ is called the voltage and the term δ|∇u| 2 is called a fringing field (cf. [START_REF] Pelesko | The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models[END_REF][START_REF] Lindsay | Asymptotics of nonlinear eigenvalue problems modeling a MEMS capacitor: Part I: Fold point asymptotics[END_REF]). The eventual singular set {x ∈ Ω, u(x) = 1} is called rupture set. When δ = 0, problem (E λ ) becomes (S λ )

-∆u = λ (1 -u) 2 in Ω, u = 0 on ∂Ω.

Recently there have been many studies on (S λ ). We summarize some of the results here:

• There exists a critical number λ * > 0 such that for 0 < λ < λ * problem (S λ ) has a minimal stable solution u λ , while for λ > λ * there are no solutions to (S λ ) (see [START_REF] Ghoussoub | On the partial differential equations of electrostatic MEMS devices: stationary case[END_REF]).

• Either the solution branch stops at λ * and lim λ→λ * u λ ∞ = 1 (if Ω is a ball in R n with n ≥ 8 for example); or the solution branch bends back, we could have another critical parameter 0 < λ * < λ * (when Ω is a ball in R n with 2 ≤ n ≤ 7; or convex domain with two axes of symmetry in R 2 ) such that the solution branch takes infinitely many turns and converges to a rupture solution of (S λ * ) (see [START_REF] Esposito | Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity[END_REF][START_REF] Guo | Asymptotic Behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity[END_REF][START_REF] Guo | Infinitely many turning points for an elliptic problem with a singular nonlinearity[END_REF]).

• For general strictly convex domains with n ≥ 2, it can be shown that for λ > 0 small, the minimal solution is the unique one for (S λ ) (see [START_REF] Esposito | Uniqueness of solutions for an elliptic equation modeling MEMS[END_REF][START_REF] Schaaf | Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry[END_REF]). So we must have a family of solutions (u k , λ k ) such that lim k→∞ λ k = λ > 0 and lim k→∞ u k ∞ = 1.

In this short note, we show that the fringing field dramatically changes the structure of solutions of (E λ ) (see Theorem 5 below): we prove that there exists a critical parameter λ * δ such that for λ > λ * δ there are no solutions to (E λ ); for 0 < λ < λ * δ there are at least two solutions; and when λ = λ * δ there exists a unique solution. Furthermore, for any fixed µ > 0, all solutions to (E λ ) with λ ≥ µ are below C µ < 1, i.e. no ruptures can occur by using solutions with λ tending to some λ > 0. Our study holds for any dimension and confirms the numerical results obtained in [START_REF] Pelesko | The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models[END_REF][START_REF] Lindsay | Asymptotics of nonlinear eigenvalue problems modeling a MEMS capacitor: Part I: Fold point asymptotics[END_REF]. Here all solutions considered are classical solutions.

The results of this paper are also true for the generalized MEMS equation

(E λ,p ) -∆u = λ(1 + δ|∇u| 2 ) (1 -u) p
in Ω, u = 0 on ∂Ω where p > 1.

A Key Transformation

To study the structure of solutions for (E λ ), we present a suitable transformation, which leads to considering a semilinear equation. More precisely, we have

Lemma 1 Let v = ζ λ (u) = u 0 e λδ 1-s ds, ∀ u ∈ [0, 1), (1)
then u : Ω → [0, 1) is a solution (resp. supersolution, subsolution) of (E λ ) if and only if v is a solution (resp. supersolution, subsolution) for

(F λ ) -∆v = ρ λ (v), v > 0 in Ω, v = 0 on ∂Ω
where ρ λ is a smooth increasing function from R + into (0, ∞), defined by

ρ λ (v) = ξ λ • ζ -1 λ with ξ λ (u) = λe λδ 1-u (1 -u) 2 . ( 2
)
Proof. As ξ λ , ζ λ are increasing in [0, 1) and lim

u→1 -ζ λ (u) = ∞, so is ρ λ in R + . By direct calculus, v = ζ λ (u) satisfies -∆v = -e λδ 1-u ∆u - λδe λδ 1-u (1 -u) 2 |∇u| 2 ,
all conclusions are straightforward.

Otherwise, it is not difficult to prove Theorem 1 Fix δ > 0, there exists λ * δ ∈ (0, ∞) such that for any λ < λ * δ , the equation (E λ ) has a minimal solution u λ , while for any λ > λ * δ , no solution exists for (E λ ). Moreover λ → u λ is increasing for λ ∈ (0, λ * δ ).

Here the minimal solution means that for any solution u to (E λ ), we have u λ ≤ u in Ω.

Proof. The result is a direct consequence of the following claims:

(i) If (E λ ) is solvable with λ > 0, then (E λ ) is solvable for any λ ∈ (0, λ).
(ii) The equation (E λ ) has no solution for λ sufficiently large.

(iii) For λ > 0 small enough, we have a solution to (E λ ).

(iv) If (E λ ) is solvable, then there exists a minimal solution u λ .

If u is a solution to (E λ ), it is clearly a supersolution to (E λ ), so v = ζ λ (u) is a supersolution to (F λ ) by Lemma 1. As ρ λ (0) = λ e λ δ > 0, 0 is always a subsolution. Moreover ρ λ is locally Lipschitz in R + , so we have a solution to (F λ ), which yields the claim (i).

The claim (ii) comes from the fact that any solution of (E λ ) is a supersolution for the equation (S λ ), which has no solution for large λ. Let -∆ξ = 1 in Ω and ξ = 0 on ∂Ω, fix c > 0 such that c ξ ∞ < 1. We can check that for cξ is a supersolution of (E λ ) if λ > 0 is small enough, this leads to the claim (iii).

The last claim is due to the monotonicity of ρ λ (cf. (4) below), ζ λ and the monotone iteration for (F λ ) as -∆v n+1 = ρ λ (v n ) with Dirichlet boundary condition and v 0 ≡ 0.

Remark 1 Of course, the transformation v = ζ λ is not really necessary for the above proof. Thanks to the monotonicity of function g(u) = (1 -u) -2 , we can consider directly the following iteration operator w = T u, the unique solution of

-∆w = λ(1 + δ|∇u| 2 ) (1 -u) 2
in Ω, w = 0 on ∂Ω.

Stability of Minimal Solutions

The minimal solution for (E λ ) will ensure some stability properties, even though the equation (E λ ) does not have a variational structure. First, for the linearized operator of (E λ ): 2 , we can define the principal eigenvalue µ 1 of L λ , associated to the Dirichlet boundary condition (cf. [START_REF] López-Gomez | The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems[END_REF]). Then a solution u of (E λ ) is said to be stable if and only if µ 1 (L λ ) ≥ 0. Another idea is to use the transformation v = ζ λ (u) and the corresponding linearized operator. Following the ideas in [START_REF] Brezis | Blow-up for u t -∆u = g(u) revisited[END_REF], we obtain

L λ ϕ = -∆ϕ - 2λ(1 + δ|∇u| 2 ) (1 -u) 3 ϕ - 2λδ∇u∇ϕ (1 -u)
Theorem 2 Let λ ∈ (0, λ * δ ), the minimal solution v λ of (F λ ) satisfies Ω |∇ϕ| 2 ≥ Ω ρ λ (v λ )ϕ 2 dx, ∀ ϕ ∈ H 1 0 (Ω). (3)
Furthermore, v λ is the unique solution of (F λ ) verifying (3) and u λ is the unique stable solution of (E λ ).

Moreover,

u = ζ -1 λ (v) implies ρ λ (v) = ξ λ • ζ -1 λ (v) = ξ λ ζ λ • ζ -1 λ (v) = λ 2 δ (1 -u) 4 + 2λ (1 -u) 3 > 0. ( 4
)
As the minimal solution u λ of (E λ ) is just ζ -1 λ (v λ ), we conclude then Theorem 3 For λ ∈ (0, λ * δ ), the minimal solution u λ is the unique solution of (E λ ) verifying the following stability condition:

Ω |∇ϕ| 2 ≥ Ω λ 2 δ (1 -u λ ) 4 + 2λ (1 -u λ ) 3 ϕ 2 dx, ∀ ϕ ∈ H 1 0 (Ω). ( 5
)

Bifurcation and Uniform Estimate

Using the equation (F λ ) and the standard bifurcation theory of Rabinowitz (section 3 of [START_REF] Rabinowitz | Some global results for nonlinear eigenvalue problems[END_REF]), we can say that, a solution curve (λ, v) exists in R + × C(Ω), it goes from (0, 0) to the "infinity". By Theorem 1, the only possibility is that v ∞ tends to ∞. For (F λ ), when v ∞ → ∞, we show that λ must tend to 0 by the following result.

Theorem 4 For any µ > 0, there exists a constant C µ > 0 such that any solution of (F λ ) with λ ≥ µ verifies v ∞ < C µ . Consequently, there exists c µ ∈ (0, 1) such that any solution u of (E λ ) with λ ≥ µ verifies u ≤ c µ < 1.

Proof. In fact, using integration by parts, we can see that

v = ζ λ (u) ∼ (1 -u) 2 λδ e λδ 1-u as u → 1 -.
Hence for µ ∈ (0, λ * δ ) fixed, there exist positive constants C, C such that

Cv(ln v) 4 ≤ ρ λ (v) ≤ C v(ln v) 4 ∀ (λ, v) ∈ [µ, λ * δ ) × [2, ∞).
We have also the uniform estimate [START_REF] Brezis | On a class of superlinear elliptic problems[END_REF] holds and shows that there exists

ρ λ (v) ≥ Cv + µ for (λ, v) ∈ [µ, λ * δ ) × R + , the proof of Theorem 2.1 in
C µ > 0 such that v ∞ < C µ < ∞.
The conclusion for u is an immediate consequence.

An important consequence is just the uniqueness of solution for (E λ * δ ). We shall use the problem (F λ ). Now v * = lim λ→λ * δ v λ is a smooth solution for the limit problem (F λ * δ ), we claim that µ 1 -∆-ρ λ * δ (v * ) = 0. In fact, the stability of v * (in the sense of (3)) means that 4), the strong maximum principle implies that φ > 0 in Ω. Remarking also that ρ λ > 0 in R + for any λ > 0, then -∆φ -ρ λ * δ (v * )φ > 0 in Ω. By multiplying with ϕ 1 and integrating by parts, we get immediately a contradiction.

µ 1 -∆ -ρ λ * δ (v * ) ≥ 0, while the definition of λ * δ prevents to have µ 1 -∆ -ρ λ * δ (v * ) > 0. Hence we get a positive eigenfunction ϕ 1 satisfying -∆ϕ 1 -ρ λ * δ (v * )ϕ 1 = 0 in Ω and ϕ 1 = 0 on ∂Ω. If we have a solution v of (F λ * δ ) such that v = v * , we know that v ≥ v * as v ≥ v λ for any λ < λ * δ . Let φ = v -v * , so -∆φ = ρ λ * δ (v) -ρ λ * δ (v * ) ≥ 0 by (
Another consequence is that v * is a bifurcation point for the solution curve, which will continue with v ∞ tending to ∞ and the associated λ must go to zero. So we get at least two solutions to (F λ ) for any λ ∈ (0, λ * δ ). Coming back to u, we obtain the main theorem of this paper.

Theorem 5 If a family of solutions {u k } of (E λ k ) verifies lim k→∞ u k ∞ = 1, then lim k→∞ λ k = 0. Furthermore, u * = lim λ→λ * δ u λ is the unique solution of the limit equation (E λ * δ ) while for any λ ∈ (0, λ * δ ), the equation (E λ ) has at least two solutions.

Estimate of λ * δ

Here we compare λ * δ with λ * in lower dimension situation.

Theorem 6 For n < 8 and δ > 0, we have

λ * 1 + δ ∇u * 2 ∞ ≤ λ * δ ≤ λ * (6)
where λ * is the critical value for the problem (S λ ) and u * is the unique solution of (S λ * ).

Proof. As any solution of (E λ ) is supersolution of (S λ ), it is clear that λ * δ ≤ λ * . On the other hand, when n < 8, u * is a smooth function with u * ∞ < 1 (see [START_REF] Esposito | Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity[END_REF]). Obviously u * is a supersolution for (E λ ) with

λ = λ * 1 + δ ∇u * 2 ∞
, so we get the lower bound.

Therefore λ * δ = λ * + O(δ) in dimension two, this confirms somehow the formal result in [START_REF] Lindsay | Asymptotics of nonlinear eigenvalue problems modeling a MEMS capacitor: Part I: Fold point asymptotics[END_REF] (see also another bound of λ * δ in section 5 of [START_REF] Pelesko | The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models[END_REF]).

Remarks and Open Questions

As we have seen in Theorem 5, the introduction of fringing field basically destroys the infinite fold point structure of the basic membrane problem (S λ ) for any smooth domain.

There are still some interesting questions:

• Do we have some weak solutions with u ∞ = 1 for (E λ )? We turn to conjecture that no weak solution exists for the fringing field model. In fact, using Sobolev embedding and boot-strap argument, any weak solution of (F λ ) satisfying ρ λ (v) ∈ L 1 (Ω) is indeed smooth. However, if u is a just weak solutions for (E λ ), it is not clear that v = ρ λ (u) is then a weak solution for (F λ ).

• In [START_REF] Lindsay | Asymptotics of nonlinear eigenvalue problems modeling a MEMS capacitor: Part I: Fold point asymptotics[END_REF], Lindsay and Ward derived the following asymptotic behavior of λ * δ :

λ * δ = λ * -Cδ + O(δ 2 ) (7)
in the case of a unit disk or a slab in R 2 , where C > 0 is a constant depending on u * of the unit disk or slab without the fringing field. Can we prove rigorously this first order expansion (7)? A key point seems to prove a uniform upper bound for v * as δ tends to zero.

• In nice domains (disks, convex domains with two axes of symmetry in R 2 ), it has been shown that for the problem (S λ ), there exists a λ * > 0 such that the solution branch has infinitely many turns as λ crosses λ * (see [START_REF] Guo | Asymptotic Behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity[END_REF][START_REF] Guo | Infinitely many turning points for an elliptic problem with a singular nonlinearity[END_REF]). On the other hand, in the presence of fringing field, there are at most finitely many turns. What is the asymptotic behavior of the solutions near λ * as δ → 0 + ?

• It seems that there are no studies on the corresponding parabolic equation

u t -∆u = λ(1 + δ|∇u| 2 ) (1 -u) 2 . ( 8
)
What is the effect of the fringing field on (8)? Can we establish results similar to [START_REF] Brezis | Blow-up for u t -∆u = g(u) revisited[END_REF][START_REF] Ghoussoub | Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS[END_REF][START_REF] Guo | Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties[END_REF]]?

Acknowledgments The research of the first author is supported by General Research Fund from RGC of Hong Kong. The second author is supported by the French ANR project referenced ANR-08-BLAN-0335-01. We both thank the Department of mathematics, East China Normal University for its kind hospitality. The first author thanks Prof. M. Ward for introducing this problem and Prof. Z.M. Guo for useful discussions.