Vitali Vougalter
email: vitali@math.toronto.edu

Vitaly Volpert
email: volpert@math.univ-lyon1.fr

SOLVABILITY IN THE SENSE OF SEQUENCES FOR SOME NON FREDHOLM OPERATORS RELATED TO THE SUPERDIFFUSION

Keywords: AMS Subject Classification: 35J10, 35P10, 47F05 solvability conditions, non Fredholm operators, Sobolev spaces

.

Introduction

Let us consider the problem

-∆ + V (x)u -au = f (1.1) with u ∈ E = H 1 (R d ) and f ∈ F = L 2 (R d ), d ∈ N, a is a constant and V (x)
is a function decaying to 0 at infinity. The operator -∆ + V (x) can be defined via the spectral calculus under the appropriate technical conditions on the scalar potential V (x) (see Assumption 3 below). If a ≥ 0, then the essential spectrum of the operator A : E → F corresponding to the left side of problem (1.1) contains the origin. Consequently, such operator fails to satisfy the Fredholm property. Its image is not closed, for d > 1 the dimensions of its kernel and the codimension of its image are not finite. In the present article we will study some properties of the operators of this kind. Let us recall that elliptic equations with non-Fredholm operators were treated extensively in recent years (see [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF], [START_REF] Vougalter | Solvability conditions for some non-Fredholm operators[END_REF], [START_REF] Vougalter | On the solvability conditions for some non Fredholm operators[END_REF], [START_REF] Vougalter | On the solvability conditions for the diffusion equation with convection terms[END_REF], [START_REF] Vougalter | Solvability relations for some non Fredholm operators[END_REF], [START_REF] Volpert | On the solvability conditions for a linearized Cahn-Hilliard equation[END_REF], [START_REF] Vougalter | Solvability conditions for a linearized Cahn-Hilliard equation of sixth order[END_REF], [START_REF] Vougalter | Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems[END_REF], also [START_REF] Benkirane | Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans R n[END_REF]) along with their potential applications to the theory of reaction-diffusion equations (see [START_REF] Ducrot | Systemes de réaction-diffusion sans propriété de Fredholm[END_REF], [START_REF] Ducrot | Reaction-diffusion problems with non Fredholm operators[END_REF]). In the particular case when a = 0 the operator A 2 satisfies the Fredholm property in some properly chosen weighted spaces [START_REF] Amrouche | Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces[END_REF], [START_REF] Amrouche | Mixed exterior Laplace's problem[END_REF], [START_REF] Bolley | Propriété d'indice en théorie Holderienne pour des opérateurs différentiels elliptiques dans R n[END_REF], [START_REF] Bolley | Propriété d'indice en théorie Hölderienne pour le problème extérieur de Dirichlet[END_REF], [START_REF] Benkirane | Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans R n[END_REF]. However, the case with a = 0 is significantly different and the approach developed in these articles cannot be adopted. One of the important questions concerning problems with non-Fredholm operators is their solvability. Let us address it in the following setting. Let f n be a sequence of functions in the image of the operator A, such that f n → f in L 2 (R d ) as n → ∞. Denote by u n a sequence of functions from H 1 (R d ) such that

Au n = f n , n ∈ N.
Because the operator A fails to satisfy the Fredholm property, the sequence u n may not be convergent. Let us call a sequence u n such that Au n → f a solution in the sense of sequences of problem Au = f (see [START_REF] Volpert | Fredholm theory of elliptic problems in unbounded domains[END_REF]). If such sequence converges to a function u 0 in the norm of the space E, then u 0 is a solution of this problem. Solution in the sense of sequences is equivalent in this sense to the usual solution. However, in the case of the operators without Fredholm property, this convergence may not hold or it can occur in some weaker sense. In such case, solution in the sense of sequences may not imply the existence of the usual solution. In the present article we will find sufficient conditions of equivalence of solutions in the sense of sequences and the usual solutions. In the other words, the conditions on sequences f n under which the corresponding sequences u n are strongly convergent. Solvability in the sense of sequences for the sums of non Fredholm Schrödinger type operators was treated in [START_REF] Vougalter | On the solvability in the sense of sequences for some non-Fredholm operators[END_REF].

In the first part of the work we consider the problem

√ -∆u -au = f (x), x ∈ R d , d ∈ N, (1.2)
where a ≥ 0 is a constant and the right side is square integrable. The operator √ -∆ here is actively used, for instance in the studies of the superdiffusion problems (see e.g. [START_REF] Vougalter | Existence of stationary solutions for some integrodifferential equations with superdiffusion[END_REF] and the references therein). Superdiffusion can be described as a random process of particle motion characterized by the probability density distribution of jump length. The moments of this density distribution are finite in the case of normal diffusion, but this is not the case for superdiffusion. Asymptotic behavior at infinity of the probability density function determines the value of the power of the Laplacian (see [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]). Another application of the √ -∆ operator is related to the relativistic Quantum Mechanics (see e.g. [START_REF] Vougalter | Sharp semiclassical bounds for the moments of eigenvalues for some Schrödinger type operators with unbounded potentials[END_REF]). The equation analogous to (1.2) but with the standard Laplacian in the context of the solvability in the sense of sequences was treated in [START_REF] Volpert | Solvability in the sense of sequences to some non-Fredholm operators[END_REF]. Note that for the operator

√ -∆ -a : H 1 (R d ) → L 2 (R d ) the essential spectrum fills the semi-axis [-a, ∞) such that its inverse from L 2 (R d ) to H 1 (R d ) is unbounded.
We write down the corresponding sequence of equations with n ∈ N as

√ -∆u n -au n = f n (x), x ∈ R d , d ∈ N, (1.3)
with the right sides convergent to the right side of (1.2) in L 2 (R d ) as n → ∞. The inner product of two functions

(f (x), g(x)) L 2 (R d ) := R d f (x)ḡ(x)dx, (1.4)
with a slight abuse of notations when these functions fail to be square integrable.

Indeed, if f (x) ∈ L 1 (R d ) and g(x) ∈ L ∞ (R d ),
then obviously the integral in the right side of (1.4) makes sense, like for example in the case of functions involved in the orthogonality conditions of Theorems 1 and 2 below. We will use the space

H 1 (R d ) equipped with the norm u 2 H 1 (R d ) := u 2 L 2 (R d ) + ∇u 2 L 2 (R d ) . (1.5)
Throughout the article, the sphere of radius r > 0 in R d centered at the origin will be denoted by S d r . The notation B d will stand for the unit ball in the space of d dimensions with the center at the origin and |B d | for its Lebesgue measure. First of all, let us formulate the solvability conditions for equation (1.2).

Theorem 1. Let f (x) ∈ L 2 (R d ), d ∈ N. a) When a = 0, d = 1, 2 and in addition xf (x) ∈ L 1 (R d ), problem (1.2) admits a unique solution u(x) ∈ H 1 (R d ) if and only if (f (x), 1) L 2 (R d ) = 0 (1.6) holds. b) When a = 0, d ≥ 3 and in addition f (x) ∈ L 1 (R d ), problem (1.2) possesses a unique solution u(x) ∈ H 1 (R d ). c) When a > 0, d = 1 and in addition xf (x) ∈ L 1 (R), equation (1.2) has a unique solution u(x) ∈ H 1 (R) if and only if f (x), e ±iax √ 2π L 2 (R) = 0 (1.7) holds. d) When a > 0, d ≥ 2 and in addition xf (x) ∈ L 1 (R d ), problem (1.2) admits a unique solution u(x) ∈ H 1 (R d ) if and only if f (x), e ipx (2π) d 2 L 2 (R d ) = 0, p ∈ S d a (1.8)
holds.

Then let us turn our attention to the issue of the solvability in the sense of sequences for our equation.

Theorem 2. Let n ∈ N and f n (x) ∈ L 2 (R d ), d ∈ N, such that f n (x) → f (x) in L 2 (R d ) as n → ∞. a) When a = 0, d = 1, 2, let in addition xf n (x) ∈ L 1 (R d ), n ∈ N, such that xf n (x) → xf (x) in L 1 (R d )
as n → ∞ and the orthogonality conditions

(f n (x), 1) L 2 (R d ) = 0 (1.9)
hold for all n ∈ N. Then problems (1.2) and (1.3)

admit unique solutions u(x) ∈ H 1 (R d ) and u n (x) ∈ H 1 (R d ) respectively, such that u n (x) → u(x) in H 1 (R d ) as n → ∞. b) When a = 0, d ≥ 3, let in addition f n (x) ∈ L 1 (R d ), n ∈ N, such that f n (x) → f (x) in L 1 (R d ) as n → ∞. Then equations (1.2) and (1.3) possess unique solutions u(x) ∈ H 1 (R d ) and u n (x) ∈ H 1 (R d ) respectively, such that u n (x) → u(x) in H 1 (R d ) as n → ∞. c) When a > 0, d = 1, let in addition xf n (x) ∈ L 1 (R), n ∈ N, such that xf n (x) → xf (x) in L 1 (R)
as n → ∞ and the orthogonality conditions

f n (x), e ±iax √ 2π L 2 (R) = 0 (1.10)
hold for all n ∈ N. Then equations (1.2) and (1.3) have unique solutions u(x) ∈ H 1 (R) and

u n (x) ∈ H 1 (R) respectively, such that u n (x) → u(x) in H 1 (R) as n → ∞. d) When a > 0, d ≥ 2, let in addition xf n (x) ∈ L 1 (R d ), n ∈ N, such that xf n (x) → xf (x) in L 1 (R d )
as n → ∞ and the orthogonality conditions

f n (x), e ipx (2π)
d 2 L 2 (R d ) = 0, p ∈ S d a (1.11)
hold for all n ∈ N. Then equations (1.2) and (1.3)

admit unique solutions u(x) ∈ H 1 (R d ) and u n (x) ∈ H 1 (R d ) respectively, such that u n (x) → u(x) in H 1 (R d ) as n → ∞.
Remark. In the statement b) of Theorem 2 our operator A has a right inverse. However, since L 1 (R d ) ∩ L 2 (R d ) is not a Hilbert space, the classical theory of operators acting in Hilbert spaces (see e.g. [START_REF] Hislop | Introduction to Spectral Theory with Applications to Schrödinger Operators[END_REF]) is not applicable here. The inverse image of the operator may not be closed in H 1 (R d ). Similar assertions can be made for analogous statements of the other theorems. On the other hand, similarly to the Laplace operator (see e.g. [START_REF] Amrouche | Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces[END_REF]- [START_REF] Benkirane | Propriété d'indice en théorie Holderienne pour des opérateurs elliptiques dans R n[END_REF]), it is possible that the operator under consideration satisfies the Fredholm property in the appropriate weighted spaces. This is not yet proved for the fractional operators.

Note that when a = 0 and the dimension of the problem is at least three, orthogonality relations in the theorems above are not required.

Let us use the hat symbol to designate the standard Fourier transform

f (p) := 1 (2π) d 2 R d f (x)e -ipx dx, p ∈ R d , d ∈ N, (1.12)
such that

f (p) L ∞ (R d ) ≤ 1 (2π) d 2 f (x) L 1 (R d ) .
(1.13)

In the second part of the article we treat the problem

-∆ + V (x)u -au = f (x), x ∈ R 3 , a ≥ 0, (1.14)
where the right side is square integrable. The correspondent sequence of equations for n ∈ N will be

-∆ + V (x)u n -au n = f n (x), x ∈ R 3 , a ≥ 0, (1.15)
where the right sides converge to the right side of (1.14

) in L 2 (R 3 ) as n → ∞.
We make the following technical assumptions on the scalar potential involved in equations above. Note that the conditions on V (x), which is shallow and shortrange will be analogous to those stated in Assumption 1.1 of [START_REF] Vougalter | Solvability conditions for some non-Fredholm operators[END_REF] (see also [START_REF] Vougalter | On the solvability conditions for some non Fredholm operators[END_REF], [START_REF] Vougalter | On the solvability conditions for the diffusion equation with convection terms[END_REF]). The essential spectrum of such a Schrödinger operator fills the nonnegative semi-axis (see e.g. [START_REF] Jonsson | [END_REF]).

Assumption 3. The potential function V (x) : R 3 → R satisfies the estimate

|V (x)| ≤ C 1 + |x| 3.5+δ
with some δ > 0 and x = (x 1 , x 2 , x 3 ) ∈ R 3 a.e. such that

4 1 9 9 8 (4π) -2 3 V 1 9 L ∞ (R 3 ) V 8 9 L 4 3 (R 3 ) < 1 and √ c HLS V L 3 2 (R 3 ) < 4π. (1.16)
Here and below C will denote a finite positive constant and c HLS given on p.98 of [START_REF] Lieb | Loss, Analysis[END_REF] is the constant in the Hardy-Littlewood-Sobolev inequality

R 3 R 3 f 1 (x)f 1 (y) |x -y| 2 dxdy ≤ c HLS f 1 2 L 3 2 (R 3 ) , f 1 ∈ L 3 2 (R 3 ).
By means of Lemma 2.3 of [START_REF] Vougalter | Solvability conditions for some non-Fredholm operators[END_REF], under Assumption 3 above on the potential function, the operator -∆ + V (x) on L 2 (R 3 ) is self-adjoint and unitarily equivalent to -∆ via the wave operators (see [START_REF] Kato | Wave operators and similarity for some non-selfadjoint operators[END_REF], [START_REF] Rodnianski | Time decay for solutions of Schrödinger equations with rough and time-dependent potentials[END_REF])

Ω ± := s -lim t→∓∞ e it(-∆+V ) e it∆ ,
where the limit is understood in the strong L 2 sense (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics, III: Scattering Theory[END_REF] p.34, [START_REF] Cycon | Schrödinger Operators with Application to Quantum Mechanics and Global Geometry[END_REF] p.90).

Thus -∆ + V (x) : H 1 (R 3 ) → L 2 (R 3 ) has only the essential spectrum σ ess ( -∆ + V (x) -a) = [-a, ∞)
and no nontrivial L 2 (R 3 ) eigenfunctions. Its functions of the continuous spectrum satisfy

-∆ + V (x)ϕ k (x) = |k|ϕ k (x), k ∈ R 3 , (1.17)
in the integral formulation the Lippmann-Schwinger equation for the perturbed plane waves (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics, III: Scattering Theory[END_REF] p.98)

ϕ k (x) = e ikx (2π) 3 2 - 1 4π R 3 e i|k||x-y| |x -y| (V ϕ k )(y)dy (1.18)
and the orthogonality relations

(ϕ k (x), ϕ q (x)) L 2 (R 3 ) = δ(k -q), k, q ∈ R 3 . (1.19)
In particular, when the vector k = 0, we have ϕ 0 (x). We designate the generalized Fourier transform with respect to these functions using the tilde symbol as

f (k) := (f (x), ϕ k (x)) L 2 (R 3 ) , k ∈ R 3 . (1.20) (1.20
) is a unitary transform on L 2 (R 3 ). The integral operator involved in (1.18) is being designated as

(Qϕ)(x) := - 1 4π R 3 e i|k||x-y| |x -y| (V ϕ)(y)dy, ϕ ∈ L ∞ (R 3 ). We consider Q : L ∞ (R 3 ) → L ∞ (R 3
). Under Assumption 3, according to Lemma 2.1 of [START_REF] Vougalter | Solvability conditions for some non-Fredholm operators[END_REF] the operator norm Q ∞ is bounded above by the quantity I(V ), which is the left side of the first inequality in (1.16), such that I(V ) < 1. Corollary 2.2 of [START_REF] Vougalter | Solvability conditions for some non-Fredholm operators[END_REF] under our assumptions gives us the estimate

| f (k)| ≤ 1 (2π) 3 2 1 1 -I(V ) f (x) L 1 (R 3 ) . (1.21)
We have the following statement concerning the solvability of problem (1.14).

Theorem 4. Let Assumption 3 hold and f

(x) ∈ L 2 (R 3 ). a) When a = 0, let in addition f (x) ∈ L 1 (R 3 ). Then equation (1.14) admits a unique solution u(x) ∈ H 1 (R 3 ). b) When a > 0, let in addition xf (x) ∈ L 1 (R 3 ). Then equation (1.14) has a unique solution u(x) ∈ H 1 (R 3 ) if and only if (f (x), ϕ k (x)) L 2 (R 3 ) = 0, k ∈ S 3 a (1.22) holds.
Our final main proposition deals with the solvability in the sense of sequences of equation (1.14). (1.14) and (1.15)

Theorem 5. Let Assumption 3 hold, n ∈ N and f n (x) ∈ L 2 (R 3 ), such that f n (x) → f (x) in L 2 (R 3 ) as n → ∞. a) When a = 0 let in addition f n (x) ∈ L 1 (R 3 ), n ∈ N, such that f n (x) → f (x) in L 1 (R 3 ) as n → ∞. Then problems
possess unique solutions u(x) ∈ H 1 (R 3 ) and u n (x) ∈ H 1 (R 3 ) respectively, such that u n (x) → u(x) in H 1 (R 3 ) as n → ∞. b) When a > 0 let in addition xf n (x) ∈ L 1 (R 3 ), n ∈ N, such that xf n (x) → xf (x) in L 1 (R 3 ) as n → ∞ and the orthogonality conditions (f n (x), ϕ k (x)) L 2 (R 3 ) = 0, k ∈ S 3 a (1.23)
hold for all n ∈ N. Then problems (1.14) and (1.15)

possess unique solutions u(x) ∈ H 1 (R 3 ) and u n (x) ∈ H 1 (R 3 ) respectively, such that u n (x) → u(x) in H 1 (R 3 ) as n → ∞.
Note that (1.22) and (1.23) are the orthogonality conditions to the functions of the continuous spectrum of our Schrödinger operator, as distinct from the Limiting Absorption Principle in which one needs to orthogonalize to the standard Fourier harmonics (see e.g. Lemma 2.3 and Proposition 2.4 of [START_REF] Goldberg | A limiting absorption principle for the threedimensional Schrödinger equation with L p potentials[END_REF]).

Solvability in the sense of sequences in the no potential case

Proof of Theorem 1. Note that the statements a) and b) of the theorem are the results of Lemma 3.1 of [START_REF] Vougalter | Existence of stationary solutions for some integrodifferential equations with superdiffusion[END_REF].

Evidently, if u(x) ∈ L 2 (R d ) is a solution of (1.
2) with a square integrable right side, it belongs to H 1 (R d ) as well. Indeed, directly from (1.2) we have

√ -∆u(x) ∈ L 2 (R d ), such that due to the fact that √ -∆u L 2 (R d ) = ∇u L 2 (R d ) , we have ∇u(x) ∈ L 2 (R d ), which yields u(x) ∈ H 1 (R d ) as well.
To show the uniqueness of solutions for our problem, let us suppose that (1.2) admits two square integrable solutions u 1 (x) and u 2 (x). Then their difference w(x) := u 1 (x)u 2 (x) ∈ L 2 (R d ) as well. Clearly, it satisfies the equation √ -∆w = aw.

Since the operator √ -∆ has no nontrivial square integrable eigenfunctions in the whole space, we have w(x) = 0 a.e. in R d .

Let us apply the standard Fourier transform (1.12) to both sides of equation (1.2). This yields

u(p) = f (p) |p| -a .
Let us first treat the case c) of the theorem, such that the dimension of the problem d = 1. We define the following sets on the real line

I + δ := [a -δ, a + δ], I - δ := [-a -δ, -a + δ], 0 < δ < a, (2.24)
such that

I δ := I + δ ∪ I - δ , R = I δ ∪ I c δ .
Here and further down A c ⊆ R d denotes the complement of the set A ⊆ R d . This enables us to express u(p) as the sum

f (p) |p| -a χ I c δ + f (p) |p| -a χ I + δ + f (p) |p| -a χ I - δ .
Note that χ I stands for the characteristic function of an interval I on the real line.

Obviously, we have I c δ = I c+ δ ∪ I c- δ , where

I c+ δ := I c δ ∩ R + , I c- δ := I c δ ∩ R -. (2.25)
Here R + and R -are the nonnegative and the negative semi-axes of the real line respectively. Apparently,

f (p) p -a χ I c+ δ ≤ | f (p)| δ ∈ L 2 (R).
Similarly,

f (p) -p -a χ I c- δ ≤ | f (p)| δ ∈ L 2 (R).
Let us use the representation

f (p) = f (a) + p a d f (s) ds ds.
From the definition (1.12) of the Fourier transform we easily obtain the bound by the finite quantity due to the one of our assumptions, namely

d f (p) dp ≤ 1 √ 2π xf (x) L 1 (R) .
This easily yields the upper bound

p a d f (s) ds ds p -a χ I + δ ≤ 1 √ 2π xf (x) L 1 (R) χ I + δ ∈ L 2 (R).
Apparently,

f (a) p -a χ I + δ ∈ L 2 (R)
if and only if f (a) = 0, which is equivalent to the orthogonality condition

f (x), e iax √ 2π L 2 (R) = 0.
To treat the singularity of the problem on the negative semi-axis, we use the formula

f (p) = f (-a) + p -a d f (s) ds ds.
This gives us the upper bound

p -a d f (s) ds ds -p -a χ I - δ ≤ 1 √ 2π xf (x) L 1 (R) χ I - δ ∈ L 2 (R). Evidently, f (-a) -p -a χ I - δ ∈ L 2 (R)
if and only if f (-a) = 0, which is equivalent to the orthogonality relation

f (x), e -iax √ 2π L 2 (R) = 0.
Let us complete the proof of the theorem with establishing the part d). When the dimension d ≥ 2, we introduce the set

A δ := {p ∈ R d | a -δ ≤ |p| ≤ a + δ}, 0 < δ < a (2.26)
and express

u(p) = f (p) |p| -a χ A δ + f (p) |p| -a χ A c δ .
(2.27)

Clearly, we have the upper bound

f (p) |p| -a χ A c δ ≤ | f (p)| δ ∈ L 2 (R d ).
To study the first term in the right side of (2.27), we will use the representation formula

f (p) = f (a, σ) + |p| a ∂ f (s, σ) ∂s ds.
Here and further down σ will stand for the angle variables on the sphere. By means for the definition of the Fourier transform (1.12), we have the upper bound

∂ f (p) ∂|p| ≤ xf (x) L 1 (R d ) (2π) d 2
.

The right side of this inequality is finite due to the one of our assumptions. This allows us to estimate

|p| a ∂ f (s,σ) ∂s ds |p| -a χ A δ ≤ xf (x) L 1 (R d ) (2π)
d 2 χ A δ ∈ L 2 (R d ).
The remaining term

f (a, σ) |p| -a χ A δ ∈ L 2 (R d )
if and only if f (a, σ) vanishes, which is equivalent to orthogonality condition (1.8) for the dimensions d ≥ 2.

Then we turn our attention to establishing the solvability in the sense of sequences for our problem in the no potential case.

Proof of Theorem 2. Suppose u(x) and u n (x), n ∈ N are the unique solutions of equations (1.2) and ( 1

.3) in H 1 (R d ), d ∈ N with a ≥ 0 respectively and it is known that u n (x) → u(x) in L 2 (R d ) as n → ∞. Then u n (x) → u(x) in H 1 (R d ) as n → ∞ as well. Indeed, √ -∆(u n (x) -u(x)) = a(u n (x) -u(x)) + (f n (x) -f (x)), which clearly yields √ -∆(u n (x)-u(x)) L 2 (R d ) ≤ a u n (x)-u(x) L 2 (R d ) + f n (x)-f (x) L 2 (R d ) → 0
as n → ∞ due to our assumptions. It can be easily verified that the left side of the inequality above equals to

∇(u n (x) -u(x)) L 2 (R d ) → 0, n → ∞.
If u(x) and u n (x), n ∈ N are the unique solutions of problems (1.2) and (1.3) in H 1 (R d ), d ∈ N respectively with a = 0 as in the cases a) and b) of the theorem, by applying the standard Fourier transform (1.12) we easily obtain

u n (p) -u(p) = f n (p) -f (p) |p| χ {|p|≤1} + f n (p) -f (p) |p| χ {|p|>1} .
(2.28)

Clearly, the second term in the right side of identity (2.28) can be bounded from above in the absolute value by

| f n (p) -f (p)|, such that f n (p) -f (p) |p| χ {|p|>1} L 2 (R d ) ≤ f n (x) -f (x) L 2 (R d ) → 0, n → ∞
by means of the one of our assumptions. Let us first address the case a) of the theorem when the dimension d = 1. Then via the part a) of Theorem 1, each equation (1.3) admits a unique solution

u n (x) ∈ H 1 (R), n ∈ N. Apparently, |(f (x), 1) L 2 (R) | = |(f (x) -f n (x), 1) L 2 (R) | ≤ f n (x) -f (x) L 1 (R) → 0
as n → ∞ due to the result of Lemma 6 below. Hence, (2.30)

(f (x), 1) L 2 (R) = 0 (2.
Using definition (1.12) of the standard Fourier transform, we easily estimate

d f n (p) dp - d f (p) dp ≤ 1 √ 2π xf n (x) -xf (x) L 1 (R) , (2.31)
such that expression (2.30) can be bounded from above in the absolute value by

1 √ 2π xf n (x) -xf (x) L 1 (R) χ {|p|≤1} .
Therefore,

p 0 d fn(s) ds -d f (s) ds ds |p| χ {|p|≤1} L 2 (R) ≤ 1 √ π xf n (x) -xf (x) L 1 (R) → 0
as n → ∞ due to the one of our assumptions. This implies that

u n (x) → u(x) in L 2 (R), n → ∞
when the dimension d = 1 and a = 0. In the case of the dimension d = 2 and a = 0, orthogonality condition

(f (x), 1) L 2 (R 2 ) = 0 (2.32)
can be obtained via the trivial limiting argument, analogously to (2.29). By means of the part a) of Theorem 1, equations (1.2) and (1.3) admit unique solutions u(x) ∈ H 1 (R 2 ) and u n (x) ∈ H 1 (R 2 ), n ∈ N respectively. Orthogonality relations (2.32) and (1.9) yield f (0) = 0, f n (0) = 0, n ∈ N when the dimension d = 2 and a = 0 as well. This allows us to express

f (p) = |p| 0 ∂ f (s, σ) ∂s ds, f n (p) = |p| 0 ∂ f n (s, σ) ∂s ds, n ∈ N.
Then the first term in the right side of identity (2.28) can be written as

|p| 0 ∂ fn(s,σ) ∂s -∂ f (s,σ) ∂s ds |p| χ {|p|≤1} . (2.33)
By means of definition (1.12) of the standard Fourier transform, we easily arrive at

∂ f n (p) ∂|p| - ∂ f (p) ∂|p| ≤ 1 2π xf n (x) -xf (x) L 1 (R 2 ) . (2.34)
Hence, expression (2.33) can be estimated from above in the absolute value by

1 2π xf n (x) -xf (x) L 1 (R 2 ) χ {|p|≤1} , such that |p| 0 ∂ fn(s,σ) ∂s -∂ f (s,σ) ∂s ds |p| χ {|p|≤1} L 2 (R 2 ) ≤ 1 2 √ π xf n (x) -xf (x) L 1 (R 2 ) → 0
as n → ∞ due to the one of our assumptions. Thus,

u n (x) → u(x) in L 2 (R 2 ), n → ∞
in the case of the dimension d = 2 and a = 0.

Then we proceed to the proof of the part b) of the theorem, when the dimension d ≥ 3 and a = 0. In this case, by means of the part b) of Theorem 1, equations (1.2) and (1.3) possess unique solutions u(x) and u n (x), n ∈ N respectively, belonging to H 1 (R d ). Let us estimate the first term in the right side of (2.28) in the absolute value from above using (1.13) by

f n (x) -f (x) L 1 (R d ) (2π) d 2 |p| χ {|p|≤1} , d ≥ 3,
such that its L 2 (R d ) norm can be bounded from above by

C f n (x) -f (x) L 1 (R d ) → 0, n → ∞
due to the one of our assumptions. Therefore,

u n (x) → u(x) in L 2 (R d ), n → ∞
in the case of the dimension d ≥ 3 and a = 0.

Then we turn our attention to the case c) of the theorem, when the dimension d = 1 and a > 0. Hence, due to the result of the part c) of Theorem 1, equation (1.3) admits a unique solution

u n (x) ∈ H 1 (R), n ∈ N. We have f n (x) ∈ L 1 (R), n ∈ N, such that f n (x) → f (x) in L 1 (R)
as n → ∞ as a result of Lemma 6 below. By virtue of the limiting argument, similarly to the proof of (2.29) we arrive at the orthogonality conditions

f (x), e ±iax √ 2π L 2 (R) = 0. (2.35)
Then by means of the result of the part c) of Theorem 1, equation (1.2) has a unique solution u(x) ∈ H 1 (R). By applying the standard Fourier transform (1.12) to both sides of (1.2) and (1.3), we arrive at

u(p) = f (p) |p| -a , u n (p) = f n (p) |p| -a , n ∈ N. (2.36)
This enables us to express u n (p)u(p) as

f n (p) -f (p) p -a χ I + δ + f n (p) -f (p) p -a χ I c+ δ + + f n (p) -f (p) -p -a χ I - δ + f n (p) -f (p) -p -a χ I c- δ , (2.37)
where I + δ , I - δ are given by (2.24) and I c+ δ , I c- δ are defined in (2.25). Clearly, the second term in (2.37) can be bounded from above in the absolute value by

| f n (p) -f (p)| δ , such that f n (p) -f (p) p -a χ I c+ δ L 2 (R) ≤ f n (x) -f (x) L 2 (R) δ → 0, n → ∞
due to the one of our assumptions. Similarly, the last term in (2.37) can be estimated from above in the absolute value by (2.38) Using (2.31), we estimate expression (2.38) from above in the absolute value by

| f n (p) -f (p)| δ . Hence f n (p) -f (p) -p -a χ I c- δ L 2 (R) ≤ f n (x) -f (x) L 2 (R) δ → 0, n → ∞
1 √ 2π xf n (x) -xf (x) L 1 (R) χ I + δ .
Therefore, the L 2 (R) norm of (2.38) can be bounded from above by

δ π xf n (x) -xf (x) L 1 (R) → 0, n → ∞
due to the one of our assumptions. Orthogonality relations (2.35) and (1.10) give us

f (-a) = 0, f n (-a) = 0, n ∈ N.
Hence, at the negative singularity

f (p) = p -a d f (s) ds ds, f n (p) = p -a d f n (s) ds ds, n ∈ N.
This enables us to estimate the third term in (2.37) from above in the absolute value by

1 √ 2π xf n (x) -xf (x) L 1 (R) χ I - δ .
Thus, its L 2 (R) norm can be bounded from above by

δ π xf n (x) -xf (x) L 1 (R) → 0, n → ∞
by means of the one of our assumptions. This proves that in dimension d = 1, when a > 0 we have

u n (x) → u(x) in L 2 (R), n → ∞.
Let us conclude the proof of the theorem with addressing the case d) when the dimension d ≥ 2 and a > 0. Then under the given assumptions, by means of the part d) of Theorem 1, equation (1.3) admits a unique solution

u n (x) ∈ H 1 (R d ), n ∈ N.
An easy limiting argument similar to the proof of (2.35) yields

f (x), e ipx (2π)
d 2 L 2 (R d ) = 0, p ∈ S d a .
(2.39)

Then by virtue of the part d) of Theorem 1, equation (1.2) has a unique solution u(x) ∈ H 1 (R d ). Using the Fourier transform (1.12), we easily obtain

u n (p) -u(p) = f n (p) -f (p) |p| -a χ A δ + f n (p) -f (p) |p| -a χ A c δ , (2.40)
where the set A δ is defined in (2.26). Clearly, the second term in the right side of (2.40) can be bounded from above in the absolute value by

| f n (p) -f (p)| δ . Thus, f n (p) -f (p) |p| -a χ A c δ L 2 (R d ) ≤ f n (x) -f (x) L 2 (R d ) δ → 0
as n → ∞ due to the one of our assumptions. Orthogonality conditions (2.39) and (1.11) give us f (a, σ) = 0, f n (a, σ) = 0, n ∈ N, such that

f (p) = |p| a ∂ f (s, σ) ∂s ds, f n (p) = |p| a ∂ f n (s, σ) ∂s ds, n ∈ N.
By virtue of the definition of the Fourier transform (1.12), similarly to inequalities (2.31) and (2.34) in lower dimensions, we easily derive

∂ f n (p) ∂|p| - ∂ f (p) ∂|p| ≤ 1 (2π) d 2 xf n (x) -xf (x) L 1 (R d ) .
Let us estimate the first term in the right side of (2.40) from above in the absolute value by 1 (2π)

d 2 xf n (x) -xf (x) L 1 (R d ) χ A δ .
Therefore,

f n (p) -f (p) |p| -a χ A δ L 2 (R d ) ≤ C xf n (x) -xf (x) L 1 (R d ) → 0, n → ∞
as assumed. This implies that in dimensions d ≥ 2, when a > 0 we have

u n (x) → u(x) in L 2 (R d )
as n → ∞.

Solvability in the sense of sequences with a scalar potential

Proof of Theorem 4. Note that the case a) of the theorem is the result of Lemma 3.3 of [START_REF] Vougalter | Existence of stationary solutions for some integrodifferential equations with superdiffusion[END_REF]. Then we proceed to proving the case of a > 0.

Let us note that it is sufficient to solve equation (1.14) in L 2 (R 3 ), since its square integrable solution will belong to H 1 (R 3 ) as well. Indeed, it can be easily verified that

-∆ + V (x)u 2 L 2 (R 3 ) = ∇u 2 L 2 (R 3 ) + R 3 V (x)|u(x)| 2 dx, (3.41)
where u(x) is a square integrable solutions of (1.14), the scalar potential V (x) is bounded due to Assumption 3 and f (x) ∈ L 2 (R 3 ) as assumed. Then by means of indentity (3.41) we have ∇u(x) ∈ L 2 (R 3 ) and therefore, u(x) ∈ H 1 (R 3 ).

To address the uniqueness of solutions of our problem, let us suppose that there exist both u 1 (x) and u 2 (x) which are square integrable in R 3 and satisfy (1.14). Then their difference w(x) := u 1 (x)u 2 (x) ∈ L 2 (R 3 ) is a solution of the equation

-∆ + V (x)w = aw.
Since the operator -∆ + V (x) has no nontrivial L 2 (R 3 ) eigenfunctions as discussed above, w(x) vanishes a.e. in R 3 .

We apply the generalized Fourier transform (1.20) with respect to the functions of the continuous spectrum of the Schrödinger operator to both sides of equation (1.14), which gives us

ũ(k) = f (k) |k| -a .
Let us introduce the spherical layer in the space of three dimensions as

B δ := {k ∈ R 3 | a -δ ≤ |k| ≤ a + δ}, 0 < δ < a. (3.42)
This allows us to express

ũ(k) = f (k) |k| -a χ B δ + f (k) |k| -a χ B c δ . (3.43)
The second term in the right side of (3.43) can be easily estimated from above in the absolute value by

| f (k)| δ ∈ L 2 (R 3 ),
since f (x) is square integrable as assumed. Let us use the representation

f (k) = f (a, σ) + |k| a ∂ f (s, σ) ∂s ds.
Hence, the first term in the right side of (3.43) can be written as

f (a, σ) |k| -a χ B δ + |k| a ∂ f (s,σ) ∂s ds |k| -a χ B δ . ( 3

.44)

The second term in sum (3.44) can be easily bounded above in the absolute value by

∇ k f (k) L ∞ (R 3 ) χ B δ ∈ L 2 (R 3 ).
Note that under our assumptions Then we proceed to the proof of our last main proposition concerning the solvability in the sense of sequences.

∇ k f (k) ∈ L ∞ (R
Proof of Theorem 5. First of all, let us show that if u(x) and u n (x), n ∈ N are the unique H 1 (R 3 ) solutions of (1.14) and (1.15) respectively and u n (x) → u(x) in L 2 (R 3 ) as n → ∞, then we have u n (x) → u(x) in H 1 (R 3 ) as n → ∞ as well. Indeed, from (1.14) and (1.15) we easily deduce that for a ≥ 0

-∆ + V (x)(u n (x) -u(x)) = a(u n (x) -u(x)) + (f n (x) -f (x)), n ∈ N. Hence -∆ + V (x)(u n (x) -u(x)) L 2 (R 3 ) ≤ a u n (x) -u(x) L 2 (R 3 ) + + f n (x) -f (x) L 2 (R 3 ) → 0, n → ∞ as assumed. Let us express -∆ + V (x)(u n (x) -u(x)) 2 L 2 (R 3 ) = ∇(u n (x) -u(x)) 2 L 2 (R 3 ) + + R 3 V (x)|u n (x) -u(x)| 2 dx,
where the scalar potential V (x) is bounded due to Assumption 3. Hence, in the identity above the left side and the second term in the right side tend to zero as n → ∞. This implies that ∇u

n (x) → ∇u(x) in L 2 (R 3 ) as n → ∞. Therefore, u n (x) → u(x) in H 1 (R 3 ) as n → ∞ as well.
In the case a) equations (1.14) and (1.15) admit unique solutions u(x), u n (x) ∈ H 1 (R 3 ), n ∈ N respectively by means of the part a) of Theorem 4 above. By applying the generalized Fourier transform (1.20) to both sides of problems (1.14) and (1.15), we arrive at

ũ(k) = f (k) |k| , ũn (k) = fn (k) |k| , n ∈ N. Hence ũn (k) -ũ(k) = fn (k) -f (k) |k| χ {|k|≤1} + fn (k) -f (k) |k| χ {|k|>1} . (3.45)
Clearly, the second term in the right side of (3.45) can be easily bounded from above in the absolute value by

| fn (k) -f (k)|, such that fn (k) -f (k) |k| χ {|k|>1} L 2 (R 3 ) ≤ f n (x) -f (x) L 2 (R 3 ) → 0, n → ∞
due to the one of our assumptions. Let us estimate the first term in the right side of (3.45) from above in the absolute value using (1.21) by

1 (2π) 3 2 1 1 -I(V ) f n (x) -f (x) L 1 (R 3 ) χ {|k|≤1} |k| .
Evidently, this implies that

fn (k) -f (k) |k| χ {|k|≤1} L 2 (R 3 ) ≤ 1 √ 2π 1 1 -I(V ) f n (x) -f (x) L 1 (R 3 ) → 0
as n → ∞ by means of the one of our assumptions. Therefore, u n (x) → u(x) in L 2 (R 3 ) as n → ∞ in the case when the parameter a = 0.

Then we turn our attention to the proof of the part b) of the theorem. By means of the result the part b) of Theorem 4, each equation (1.15) admits a unique solution u n (x) ∈ H 1 (R 3 ), n ∈ N. Then for k ∈ S 3 a a.e. via (1.21), we have

|(f (x), ϕ k (x)) L 2 (R 3 ) | = |(f (x) -f n (x), ϕ k (x)) L 2 (R 3 ) | ≤ ≤ 1 (2π)
∇ k [ fn (k) -f (k)] L ∞ (R 3 ) χ B δ .
Hence the L 2 (R 3 ) norm of (3.48) can be estimated from above by

C ∇ k [ fn (k) -f (k)] L ∞ (R 3 ) → 0, n → ∞
by means of Lemma 3.4 of [START_REF] Volpert | Solvability in the sense of sequences to some non-Fredholm operators[END_REF] under our assumptions. This proves that u n (x) → u(x) in L 2 (R 3 ) as n → ∞ when a > 0.

Auxiliary results

The following elementary lemma is used when establishing the solvability in the sense of sequences in the theorems above, with and without a scalar potential. To show that the part b) holds, we trivially estimate the norm using the Schwarz inequality as

Lemma 6. a) Let f (x) ∈ L 2 (R d ), d ∈ N and xf (x) ∈ L 1 (R d ). Then f (x) ∈ L 1 (R d ). b) Let n ∈ N, f n (x) ∈ L 2 (R d ), d ∈ N, such that f n (x) → f (x) in L 2 (R d ) as n → ∞. Let xf n (x) ∈ L 1 (R d ), such that xf n (x) → xf (x) in L 1 (R d ) as n → ∞. Then f n (x) → f (x) in L 1 (R d ) as n → ∞.
f n (x) -f (x) L 1 (R d ) ≤ ≤ |x|≤1 |f n (x) -f (x)| 2 dx |x|≤1 1dx + |x|>1 |x||f n (x) -f (x)|dx ≤ ≤ f n (x) -f (x) L 2 (R d ) |B d | + xf n (x) -xf (x) L 1 (R d ) → 0, n → ∞
by means of our assumptions.

  29) holds. By virtue of the part a) of Theorem 1, equation (1.2) has a unique solution u(x) ∈ H 1 (R) when the dimension d = 1 and a = 0. Orthogonality conditions (2.29) and (1.9) give us f (0) = 0, f n (0) = 0, n ∈ N in such case. This enables us to use the representations ∈ N, which allows us to write the first term in the right side of identity (2

  as assumed. Orthogonality conditions (2.35) and (1.10) yield f (a) = 0, f n (a) = 0, n ∈ N, n (s) ds ds, n ∈ N, which allows us to write the first term in (2.37) as

3 2 1 1 -∂

 321 I(V ) f n (x)f (x) L 1 (R 3 ) → 0, n → ∞by virtue of our assumptions along with Lemma 6 below. Therefore,(f (x), ϕ k (x)) L 2 (R 3 ) = 0, k ∈ S 3 a (3.46)holds. Then by virtue of the result the part b) of Theorem 4, equation (1.14) possesses a unique solution u(x) ∈ H 1 (R 3 ). Let us apply the generalized Fourier transform (1.20) to both sides of equations(1.14) and(1.15). Thus, we arrive atũn (k) -ũ(k) = fn (k) -f (k) |k|a χ B δ + fn (k) -f (k)with B δ defined in (3.42). Clearly, the second term in the right side of (3.47) can be bounded from above in the absolute value by| fn (k) -f (k)| δ , such that fn (k) -f (k) |k|a χ B c δ L 2 (R 3 ) ≤ f n (x)f (x) L 2 (R 3 ) δ → 0, n → ∞according to the one of our assumptions. By means of orthogonality conditions (1.23) and (3.46), we have f (a, σ) = 0, fn (a, σ) = 0, n ∈ N. fn (s, σ) ∂s ds, n ∈ N, such that the first term in the right side of (3.47) can be written as |k| a ∂ fn(s,σ) ∂s -∂ f (s,σ) ∂s ds |k|a χ B δ . (3.48) Expression (3.48) can be easily bounded from above in the absolute value by

Proof.

  To prove the part a), we express the norm f(x) L 1 (R d ) as |x|≤1 |f (x)|dx + |x|>1 |f (x)|dx.By applying the Schwarz inequality, this can be bounded from above by|x|≤1 |f (x)| 2 dx |x|≤1 1dx + |x|>1 |x||f (x)|dx ≤ ≤ f (x) L 2 (R d ) |B d | + xf (x) L 1 (R d ) < ∞due to our assumptions.

  3 ) by virtue of Lemma 2.4 of[START_REF] Vougalter | Solvability conditions for some non-Fredholm operators[END_REF]. The first term in (3.44) belongs to L 2 (R 3 ) if and only if f (a, σ) vanishes, which gives orthogonality condition(1.22).

Acknowledgements

The second author was partially supported by the Ministry of Education and Science of the Russian Federation (the agreement number 02.a03.21.0008).


Martine Marion
email: martine.marion@ec-lyon.fr

Vitaly Volpert

Existence of pulses for the system of competition of species

Keywords: Existence of pulses, Monotone systems, Leray-Schauder method Mathematics Subject Classification (2000) MSC 35K57, MSC 47F05

The paper is concerned with the existence of pulses for monotone reaction-diffusion systems of two equations. For a general class of systems we prove that pulses exist if and only if the wave solutions propagate at positive speed. This result is applied to investigate the existence of pulses for the system of competition of species.

Introduction

We consider the system of two equations

w 1 + F 1 (w 1 , w 2 ) = 0, w 2 + F 2 (w 1 , w 2 ) = 0, (1)
where the function F = (F 1 , F 2 ) is assumed to be sufficiently smooth and to satisfy the monotonicity condition:

∂F i ∂w j (w) > 0 for i = j and w = (w 1 , w

2 ) ∈ R 2 + , (2)
together with F 1 (0, 0) = F 2 (0, 0) = 0. We aim to study the existence of pulses, that is of positive solutions of system [START_REF] Doelman | Stability analysis of singular patterns in the 1-D Gray-Scott model[END_REF] with the limits at infinity w(±∞) = (0, 0). The existence of pulses has been proved for some reaction-diffusion systems: the Gray-Scott model [START_REF] Doelman | Stability analysis of singular patterns in the 1-D Gray-Scott model[END_REF], [START_REF] Hale | Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis[END_REF], [START_REF] Wei | Stability of spiky solutions in a reaction-diffusion system with four morphogens on the real line[END_REF], the Gierer-Meinhardt model [START_REF] Nec | The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system[END_REF], [START_REF] Wei | Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R 1[END_REF] and a three component system [START_REF] Doelman | Pulse dynamics in a three-component system: existence analysis[END_REF]. Singular perturbation methods to study existence and stability of pulses for a system of two equations are used in [START_REF] Doelman | An explicit theory for pulses in two component, singularly perturbed, reaction-diffusion equations[END_REF]. In a previous work [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] we studied the particular case where the nonlinear terms in (1) take the form:

F 1 (w 1 , w 2 ) = f 1 (w 2 ) -w 1 , F 2 (w 1 , w 2 ) = f 2 (w 1 ) -w 2 .
Note that in that case the graph w 1 = f 1 (w 2 ) (respectively w 2 = f 2 (w 1 )) corresponds to the zeros of F 1 (respectively F 2 ).

Here we aim to consider more general monotone systems. Let us state our assumptions on F . For simplicity, we consider infinitely differentiable functions. Setting w + = (0, 0) and w -= (1, 1) we suppose that F (w ± ) = (0, 0), the eigenvalues of the matrices F (w ± ) are negative.

(

) 3
Under these conditions F necessarily possesses at least one additional zero in the unit open square and we assume the uniqueness of such a zero, more precisely F has three zeros w + , w -, w = ( w1 , w2 ) in R 2 + ; furthermore 0 < wi < 1 and the matrix F ( w) has a positive eigenvalue.

(

) 4
Next we assume that the zeros of F 1 lie in the graph of some function parametrized by w 2 , more precisely:

F 1 (w 1 , w 2 ) = 0, w 1 , w 2 ≥ 0, ⇔ w 1 = f 1 (w 2 ), (5)
where f 1 (0) = 0, f 1 (1) = 1, f 1 (w 2 ) > 0 for w 2 ≥ 0.

Similarly, we assume that the zeros of F 2 are determined by some function depending on w 1 :

F 2 (w 1 , w 2 ) = 0, w 1 , w 2 ≥ 0, ⇔ w 2 = f 2 (w 1 ), [START_REF] Volpert | Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations[END_REF] with f 2 (0) = 0, f 2 (1) = 1, f 2 (w 1 ) > 0 for w 1 ≥ 0.

Under the above conditions we aim to investigate the existence of pulses such that: w(x) = w(-x), w (x) < 0 for x > 0 (here and everywhere below inequalities for vectors mean that each component of the vectors satisfies this inequality). Then the problem on the whole axis reduces to looking for solutions of system (1) defined on the half-axis R + such that w (0) = 0, w (x) < 0 for x > 0, w(∞) = 0.

As already mentioned this problem was studied in [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] for specific nonlinear terms. We showed that the existence of pulses is related to the sign of the wave speed for the traveling-wave solution of the reaction-diffusion system

∂v ∂t = ∂ 2 v ∂x 2 + F (v). ( 10
)
This solution reads v(x, t) = u(x-ct) where u satisfies the system of equations u + cu + F (u) = 0 [START_REF] Volpert | Elliptic partial differential equations[END_REF] and has the limits at infinity

u(±∞) = w ± . ( 12
)
Under the assumptions ( 2)-( 4), it is classical that there exists a uniquely defined traveling-wave (up to some translation in space for u). Moreover, it is a monotonically decreasing vector-function (component-wise) (see [START_REF] Volpert | Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations[END_REF], [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]).

The main result of this work is given by the following theorem.

Theorem 1.1. Under the assumptions ( 2)-( 8), Problem (1) has a solution satisfying [START_REF] Volpert | Spectrum of elliptic operators and stability of travelling waves[END_REF] if and only if the wave speed c in ( 11)-( 12) is positive.

This result will be applied to investigate the existence of pulses for the system of competition of species. In order to apply the general existence result we will need to introduce some suitable perturbed system. We also discuss conditions which guarantee the positiveness of the speed.

The paper is organized as follows. In Section 2 we first recall some results concerning the functional setting for Problem [START_REF] Doelman | Stability analysis of singular patterns in the 1-D Gray-Scott model[END_REF] and some estimates of the solutions borrowed from [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF]. Then the existence part in Theorem 1.1 is derived in Section 3. The proof relies on the Leray-Schauder method. We will reduce the nonlinearity to a model problem such that F 1 = F 2 and will look for a solution with w 1 = w 2 . In order to obtain a priori estimates of solutions, we will have to verify that the wave speed remains positive in the process of homotopy. This condition will require some additional constructions. It is worth noting that the homotopy arguments are completely different from the ones in [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF]. Finally Section 4 is devoted to the system of competition of pulses.

Functional spaces and estimates of solutions

Let us consider the system depending on some parameter τ ∈ [0, 1]

w 1 + F τ 1 (w 1 , w 2 ) = 0, w 2 + F τ 2 (w 1 , w 2 ) = 0, (13)
on the half axis x > 0 with the boundary conditions and the conditions at infinity:

w i (0) = 0, w i (∞) = 0, i = 1, 2. ( 14
)
For simplicity we will suppose that the function

F τ = (F τ 1 , F τ 2 )
is infinitely differentiable with respect to both variables w ∈ R 2 and τ ∈ [0, 1]. It satisfies the following conditions that are extensions of ( 2)-( 8):

∂F τ i ∂w j (w) > 0, i, j = 1, 2, i = j, w ∈ R 2 + , τ ∈ [0, 1], (15)
F τ (w ± ) = (0, 0), the eigenvalues of the matrices (F τ ) (w ± ) are negative, (16) F τ has three zeros w + , w -, wτ = ( wτ 1 , wτ 2 ) in R 2 + ; furthermore 0 < wτ i < 1 and the matrix (F τ ) ( wτ ) has a positive eigenvalue, (17)

F τ 1 (w 1 , w 2 ) = 0 ⇔ w 1 = f τ 1 (w 2 ), f τ 1 (w 2 ) > 0 for w 2 ≥ 0, ( 18
)
F τ 2 (w 1 , w 2 ) = 0 ⇔ w 2 = f τ 2 (w 1 ), f τ 2 (w 1 ) > 0 for w 1 ≥ 0. ( 19
)

Operators and spaces

For the functional setting let us introduce the Hölder space C k+α (R + ) consisting of vector-functions of class C k , which are continuous and bounded on the half-axis R + together with their derivatives of order k, and such that the derivatives of order k satisfy the Hölder condition with the exponent α ∈ (0, 1). The norm in this space is the usual Hölder norm. Set

E 1 = {w ∈ C 2+α (R + ), w i (0) = 0, i = 1, 2}, E 2 = C α (R + ).
Next we introduce the weighted spaces E 1 µ and E 2 µ with µ(x) = √ 1 + x 2 . These spaces are equipped with the norms:

w E i µ = wµ E i , i = 1, 2.
In view of [START_REF] Wei | Stability of spiky solutions in a reaction-diffusion system with four morphogens on the real line[END_REF], let us consider the operator

A τ (w) = w + F τ (w), (20)
acting from E 1 µ into E 2 µ . Then the linearized operator about any function in E 1 µ satisfies the Fredholm property and has the zero index. The nonlinear operator is proper on closed bounded sets. This means that the inverse image of a compact set is compact in any closed bounded set in E 1 µ . Finally, the topological degree can be defined for this operator. All these properties can be found in [START_REF] Volpert | Properness and topological degree for general elliptic operators[END_REF], [START_REF] Volpert | Elliptic partial differential equations[END_REF].

Hereafter we consider solutions of ( 13)-( 14) belonging to the space E 1 µ .

A priori estimates

We aim to derive various estimates of the solutions of ( 13)- [START_REF] Wei | Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R 1[END_REF]. Note that this problem possesses the trivial solution w ≡ w + . We will be particularly interested in monotonically decreasing solutions. We start with some preliminary properties of such solutions Lemma 2.1. Under assumptions (15)-( 19), assume that w is a monotonically decreasing non trivial solution of ( 13)- [START_REF] Wei | Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R 1[END_REF]. Then for x > 0 we have w (x) < 0 and w(x) < w -. This result is a variant of similar ones in [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] (see Lemma 2.3 and the arguments in the proof of Theorem 2.1). The details are left to the reader.

Next we aim to derive estimates in E 1 µ of the monotonically decreasing solutions of ( 13)- [START_REF] Wei | Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R 1[END_REF]. They will follow from the positiveness of the wave speed for the traveling wave problem

u + cu + F τ (u) = 0. ( 21
)
In view of the assumptions (15)-( 17), there exists a unique value of c denoted by c τ such that the system (21) has a monotonically decreasing solution with the limits at infinity:

u(±∞) = w ± . (22)
The following result is proved in [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF].

Theorem 2.2. Under assumptions (15)-( 19), moreover suppose that c τ > 0 for all τ ∈ [0, 1]. Then there exists some constant R > 0 such that for all τ ∈ [0, 1] and for any arbitrary monotonically decreasing solution w ∈ E 1 µ of ( 13)-( 14) the following estimate holds :

w E 1 µ ≤ R.
Next we state a result of separation between the monotonically decreasing solutions of (13) that will be denoted by w M (x) and the non-monotone solutions of (13) that will be denoted by w N (x).

Theorem 2.3. Suppose that the assumptions (15)-(19) hold true. Then there exists a constant r > 0 such that for any monotonically decreasing solution w M and any non-monotone solution w N (and for all τ ∈ [0, 1]) the following estimates hold :

w M -w N E 1 µ ≥ r, w M 1 (0) > r, w M 2 (0) > r.
We refer to [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] for the proof of this result.

3 Proof of Theorem 1.1

The non-existence part in Theorem 1.1 for c ≤ 0 can be proved as in [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] to which the reader is referred.

Hereafter we assume c > 0 and we aim to derive the existence of a monotonically decreasing pulse. The proof relies on the Leray-Schauder method. The key step consists in the construction of a continuous deformation (homotopy) of our problem to a model problem for which we will be able to prove that the value of the topological degree is different from zero.

The continuous deformation will take the form ( 13) and we will require that the conditions (15)-( 19) are satisfied. Besides as τ varies we will need the speed c τ in ( 21)-( 22) to remain positive. Thanks to Theorem 2.2 this will yield a priori estimates in E 1 µ of the monotonically decreasing solutions.

In Section 3.1 we first describe two tools that will allow us to keep the speed c τ positive and will be used repeatedly. Next the homotopy argument is described in Section 3.2 and we conclude the proof of the existence of a pulse in Section 3.3.

Positiveness of the wave speed

Consider the problem

u + cu + G(u) = 0, u(±∞) = w ± . ( 23
)
Here G is some arbitrary function satisfying the conditions ( 2)-( 8) (with F = G and f i = g i for the zero lines). Then there is a monotone solution of problem (23) for a unique value of the speed c. Even though we keep the same notation for the speed it is of course different than the one in ( 11)- [START_REF] Volpert | Elliptic partial differential equations[END_REF]. In particular the speed for (23) is not assumed to be positive.

We aim to modify the function G so that the conditions ( 2)-( 8) remain satisfied, the zeros of the nonlinearities remain unchanged and the speed of the traveling wave for the new nonlinearity becomes positive.

For that purpose let us introduce infinitely differentiable and non-negative functions φ i (w), i = 1, 2, whose supports are defined as follows (Figure 1).

We choose a point w 0 = (w 0 1 , w 0 2 ) such that G i (w 0 ) > 0 for i = 1, 2. Due to the assumptions ( 2)-( 8) such a point necessarily belongs to the zone between the two lines of zeros in the upper part of the square. Also note that where 1 and 2 are positive numbers and sufficiently small so that G 1 (w) > 0 for w ∈ D 1 . Similarly, we introduce

G 1 (w 0 1 , w 2 ) > 0 for w 2 ≥ w 0 2 and G 2 (w 1 , w 0 2 ) > 0 for w 1 ≥ w 0 1 . Let us set D 1 = {w = (w 1 , w 2 ), w 0 1 -1 ≤ w 1 ≤ w 0 1 + 1 , w 0 2 -2 ≤ w 2 ≤ 1},
D 2 = {w = (w 1 , w 2 ), w 0 2 -δ 1 ≤ w 2 ≤ w 0 2 + δ 1 , w 0 1 -δ 2 ≤ w 1 ≤ 1},
where δ 1 and δ 2 are sufficiently small positive numbers so that G 2 (w) > 0 for w ∈ D 2 . Next, we can introduce two rectangles D 0 i , i = 1, 2, with common width σ located inside the rectangles D i as it is shown in Figure 1 (right). We define the functions φ i in such a way that the following conditions are satisfied:

∂φ 1 ∂w 2 ≥ 0, ∂φ 2 ∂w 1 ≥ 0, (24)
and

φ i ≥ 0, Supp φ i ⊂ D i , φ i (w) = 1 for w ∈ D 0 i , i = 1, 2. ( 25
)
We now consider the nonlinearities:

H i (w) = G i (w) + kφ i (w), i = 1, 2, ( 26
)
where k is some positive constant. The function H i , i = 1, 2, differs from G i in a zone where G i > 0 (see Figure 1) and it is obtained by adding some positive term. Consequently the zeros of H i coincide with the ones of G i and H i (w) > 0 at any point w such that G i (w) > 0. The conditions (3)-( 8) remain valid for H and the monotonicity condition

∂H i ∂w j (w) > 0, i = j, (27)
is satisfied.

Hence the traveling wave problem for this modified nonlinearity which reads:

u + cu + G(u) + kφ(u) = 0, u(±∞) = w ± , (28)
where we set φ(u) = (φ 1 (u), φ 2 (u)) has a monotone solution for a unique value of the speed. This value depends on k and we denote it by c k . We aim to show that if k is large enough then c k > 0.

Lemma 3.1. Let G be a function satisfying the conditions ( 2)-( 8) and let w 0 such that G(w 0 ) > 0. Define the rectangles D, D 0 i (with width σ) and the function φ i as above. Then if

k > G ∞ σ ( 29
)
the speed c k in (28) is positive.

Proof. Let u(x) = (u 1 (x), u 2 (x)
) denote the solution of (28). Since both components of the solution are monotone functions of x, we can express one coordinate as a function of the other one: either u 2 = p(u 1 ) or u 1 = q(u 2 ). Both of them are monotone functions of their arguments.

Next consider the solution u(x) as a curve in the plane (w 1 , w 2 ). It connects the points w + and w -. Therefore it crosses the set 2, right). Moreover necessarily the curve crosses the whole width of either D 0 1 or D 0 2 .

D 0 = D 0 1 ∪ D 0 2 (Figure
Without loss of generality we can suppose that the curve crosses the whole width of D 0 1 . Then let us consider the function u 2 = p(u 1 ) along the curve. The system (28) reduces to the single equation

u 1 + c k u 1 + G 1 (u 1 , p(u 1 )) + kφ 1 (u 1 , p(u 1 )) = 0. ( 30
)
For this scalar equation it is easy to see that

c k = 1 u 1 2 L 2 (-∞,∞) 1 0 [G 1 (s, p(s)) + kφ 1 (s, p(s))] ds. Hence c k > 0 if and only if 1 0 [G 1 (s, p(s)) + kφ 1 (s, p(s))] ds > 0. ( 31
)
Since the function φ 1 is non-negative and φ 1 = 1 on D 0 1 , we have

1 0 φ 1 (s, p(s))ds > σ,
where σ is the width of the rectangle D 0 1 . Consequently the inequality (31) is satisfied if the stronger condition

1 0 G 1 (s, p(s))ds + kσ > 0 (32)
holds true. This last condition is clearly satisfied under the assumption (29) (where G ∞ denotes the supremum norm of G on the unit square). This implies that c k > 0 and concludes the proof of the lemma.

We will also make use of a classical result that states that the speed of the waves increases if the nonlinearity is increased (see [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]).

Theorem 3.2. Consider the two problems

u + cu + G k (u) = 0, u(±∞) = w ± , k = 1, 2, ( 33
)
where the functions

G k = (G k 1 , G k 2 )
are assumed to satisfy conditions (2) (possibly non-strict) and ( 3), and furthermore

G 1 i (u) ≥ G 2 i (u) for w + ≤ u ≤ w -and i = 1, 2.
If the two problems (33) possess monotone solutions with the values of the speed c = c k , k = 1, 2, then we have c 1 ≥ c 2 .

Homotopy

We aim now to construct the homotopy F τ . It is of course such that, for τ = 0, F 0 = F the given nonlinearity in (1) for which we recall that the wave speed c is assumed to be positive.

As already noted we need the speed c τ for the traveling wave problem ( 21)-( 22) to remain positive in the homotopy process. For that purpose we introduce some (arbitrary) point w 0 ∈ (0, 1) 2 such that F (w 0 ) > 0 and consider the function φ = (φ 1 , φ 2 ) constructed as above in Section 3.1 (with G = F ). We will require the homotopy to be such that F τ (w 0 ) remains positive as τ varies so that Lemma 3.1 will apply with G = F τ and φ defined as above.

Step 1 (τ ∈ [0, τ 1 ], τ 1 = 1/4). We start by increasing the function F and introduce:

F τ i (w) = F i (w) + τ Kφ i (w), i = 1, 2, 0 ≤ τ ≤ τ 1 . ( 34
)
Here K is some arbitrary positive constant which will be specified below.

As noted in Section 3.1 the functions F τ i satisfy the conditions (15)-( 19) and there zero lines coincide with the ones of F i , that is w 1 = f 1 (w 2 ) and w 2 = f 2 (w 1 ).

Let c τ denote the value of the speed for the problem

u + c τ u + F τ (u) = 0, u(±∞) = w ± . ( 35
)
Since F τ i ≥ F i , Theorem 3.2 yields that c τ ≥ c > 0 for τ ∈ [0, τ 1 ] (without any assumption on K).

Step 2 (τ ∈ [τ 1 , τ 2 ], τ 2 = 1/2). Let us introduce the functions:

F1 (w) = f 1 (w 2 ) -w 1 , F2 (w) = f 2 (w 1 ) -w 2 .
(36)

The second step of the homotopy will consist in reducing the functions F τ1 i to the functions Fi while keeping K fixed. Hence we consider

F τ i (w) = τ -τ 1 τ 2 -τ 1 Fi (w) + τ 2 -τ τ 2 -τ 1 F i (w) + τ 1 Kφ i (w), i = 1, 2, τ 1 ≤ τ ≤ τ 2 .
(37) The above functions satisfy the monotonicity condition (15) and their zero lines are still given by w 1 = f 1 (w 2 ) and w 2 = f 2 (w 1 ). Also F τ (w 0 ) > 0. Hence, due to Lemma 3.1, the wave speed c τ in (35) is positive for τ ∈ [τ 1 , τ 2 ] if K is sufficiently large.

Step 3 (τ ∈ [τ 2 , τ 3 ], τ 3 = 3/4). At the end of the step 2 we obtained

F τ2 1 (w) = f 1 (w 2 )-w 1 +τ 1 Kφ 1 (w), F τ2 2 (w) = f 2 (w 1 )-w 2 +τ 1 Kφ 2 (w). ( 38
)
We now aim to reduce the two functions f 1 and f 2 in (38) to some common value f 0 while keeping the other terms unchanged. Hence the homotopy will take the form

F τ 1 (w) = f τ 1 (w 2 ) -w 1 + τ 1 Kφ 1 (w), (39)
F τ 2 (w) = f τ 2 (w 1 ) -w 2 + τ 1 Kφ 2 (w), τ 2 ≤ τ ≤ τ 3 , where f τ2 i = f i and f τ3 1 = f τ3 2 = f 0 .
In order to satisfy the conditions (15)-(17) we will require that

f τ i (s) > 0 for s ≥ 0, i = 1, 2. ( 40
)
The system w 1 = f τ 1 (w 2 ), w 2 = f τ 2 (w 1 ) has three solutions w + , w -, wτ in R 2 + , furthermore 0 < wτ i < 1.

(41)

f τ 1 (0)f τ 2 (0) < 1, f τ 1 (1)f τ 2 (1) < 1, f τ 1 ( wτ 2 )f τ 2 ( wτ 1 ) > 1. ( 42
)
These conditions yield that the graphs of the two functions f τ 1 and f τ 2 are located with respect to each other as represented in Figure 1. Besides we ask that

F τ (w 0 ) > 0, (43)
so that the speed of the wave in (35) is positive if K is sufficiently large. Finally in preparation of the step 4 below we require that the function

f 0 = f τ3 1 = f τ3 2 moreover satisfies
The equation f 0 (s) = s has three solutions : 0, 1, µ;

0 < µ < 1, ( 44
) 1 0 (f 0 (s) -s)ds > 0. ( 45
)
We claim that we can define the functions f τ i in such a way that all the above conditions are satisfied. Let us describe this homotopy in more details.

Recall that the system of equations w 1 = f 1 (w 2 ), w 2 = f 2 (w 1 ), has a single solution ( w1 , w2 ) satisfying the inequalities 0 < wi < 1, i = 1, 2. Let us introduce the inverse function of f 1 denoted by f1 . Then the equation f 2 (s) = f1 (s) has a single solution w1 in the interval (0, 1). Now consider a function h 2 such that

h 2 (0) = 0, h 2 ( w1 ) = f 2 ( w1 ), h 2 (1) = 1, (46)
h 2 (s) > f 2 (s) for w1 < s < 1; h 2 (s) < f 2 (s) for 0 < s < w1 . (47)
Next for θ ∈ [0, 1] introduce the function:

h θ 2 = (1 -θ)f 2 + θh 2 .
Then the equation h θ 2 (s) = g 1 (s) has exactly three solutions in [0, 1], namely 0, 1 and w1 .

An explicit form of the function h 2 can be given by:

h 2 (s) =    s for 0 ≤ s < a, 1 (s -w1 ) for a ≤ s ≤ b, (s -1) + 1 for b ≤ s ≤ 1,
where > 0 and a, b are obtained by requiring the continuity of h 2 at these points. Then if is sufficiently small the functions f 1 and h θ 2 satisfy the conditions (40)-(43) (with f τ 1 = f 1 and f τ 2 = h θ 2 ) except at the points where h θ 2

is not differentiable but we can use smoothing procedure to make it infinitely differentiable and satisfy the conditions everywhere. The above procedure allowed to reduce the function f 2 to h 2 with the conditions (40)-(43) being satisfied. Next similarly, we reduce f 1 to the piecewise linear function h 1 with

h 1 (s) =    s for 0 ≤ s < r, 1 (s -w2 ) for r ≤ s ≤ d, (s -1) + 1 for d ≤ s ≤ 1 (up to smoothing).
At this point, in the homotopy process, the couple (f 1 , f 2 ) became (h 1 , h 2 ) which are piece-wise linear function depending on > 0, w1 and w2 (up to smoothing). Now in the definition of h 1 and h 2 we replace the point w by by α w with α ∈ [0, 1]. Then if and α are sufficiently small the conditions (40)-(43) remain valid and moreover

1 0 (h 1 (s) -s)ds > 0, 1 0 (h 2 (s) -s)ds > 0.
Note that either h 1 ≥ h 2 or h 2 ≥ h 1 . For sufficiently small and α let us consider the function f 0 which is the maximum of these two functions. The homotopy of the functions h 1 to f 0 (or h 2 to f 0 ) can be easily constructed in such a way that the conditions on the homotopy are verified.

Step 4 (τ ∈ [τ 3 , 1]). In this last step, we eliminate the functions φ i by setting:

F τ i (w) = f 0 (w j ) -w i + τ 1 1 -τ 1 -τ 3 Kφ i (w), i = 1, 2 and j = i, τ 3 ≤ τ ≤ 1.
(48) Note that F τ i decreases as τ increases. Hence by Theorem 3.2 the speed of the corresponding traveling wave decreases. However it remains larger than the one for the limit problem corresponding to τ = 1. This problem reads

u + c u + F 1 (u) = 0, u(±∞) = w ± , with F 1 (u) = (f 0 (u 2 ) -u 1 , f 0 (u 1 ) -u 2 ).
It possesses a uniquely defined solution such that u 1 = u 2 . The corresponding speed c 1 has the sign of the integral in (45) which is positive. Hence for τ ∈ [τ 3 , 1] we conclude that c τ ≥ c 1 > 0.

Existence of solutions

We have constructed the homotopy from our original problem to the system

w 1 + f 0 (w 2 ) -w 1 = 0, w 2 + f 0 (w 1 ) -w 2 = 0, ( 49
)
on the half axis x > 0 with the boundary condition and the condition at infinity:

w (0) = 0, w(∞) = 0. ( 50
)
Let us look for some solution of problem ( 49)-( 50) for which w 1 ≡ w 2 so that w 1 satisfies

w 1 + f 0 (w 1 ) -w 1 = 0, w 1 (0) = 0, w 1 (∞) = 0. ( 51
)
Due to the properties of f 0 and in particular (45) it is easily seen that ( 51) possesses a unique solution satisfying w 1 (x) < 0 for x > 0.

We can now prove the existence of solutions of problem (1) satisfying [START_REF] Volpert | Spectrum of elliptic operators and stability of travelling waves[END_REF]. In Section 3.2 we constructed the homotopy F τ . Let us now consider the spaces introduced in Section 2 and the operator A τ given by (20).

Since c τ > 0 for τ ∈ [0, 1], it satisfies the conditions imposed to obtain a priori estimates of monotonically decreasing solutions (Theorem 2.2). Denote by B ⊂ E 1 µ a ball which contains all monotonically decreasing solutions. Since the operator A τ is proper on closed bounded sets with respect to both variables w and τ , the set of monotonically decreasing solutions of the problem A τ (w) = 0 is compact. Since they are separated from non-monotone solutions (Theorem 2.3), then we can construct a domain D ⊂ B ⊂ E 1 µ such that all monotonically decreasing solutions (for all τ ∈ [0, 1]) are located inside D and there are no other solutions in the closure D. Indeed it is sufficient to take a union of small balls of the radius r (Theorem 2.3) around each monotonically decreasing solution.

Let us note that only strictly monotone solutions belong to this domain. In particular, the trivial solution w ≡ 0 does not belong to it since the monotonically decreasing solutions are separated from the trivial solution (Theorem 2.3). This is an important remark because the index of the trivial solution equals one. If it belongs to the domain D, then the sum of the indices, which equals the value of the degree, can be zero (see below). If it was the case, then we could not affirm the existence of solutions for the original problem.

We can now determine the topological degree τ , D). We have

γ(A 0 , D) = γ(A 1 , D) = j ind u j ,
where the sum is taken with respect to all monotonically decreasing solutions u j of problem ( 49)-( 50), and ind u j denotes the index of the solution. We have ind u j = (-1) νj , where ν j is the number of positive eigenvalues of the linearized problem. We proved in [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] that ν j = 1 so that ind u j = -1. Hence the degree is different from zero for the original problem, and the equation A 0 (w) = 0 has a solution in D. This concludes the proof of the existence result in Theorem 1.1 Let us note that we used the existence of solutions of the model problem but not their uniqueness. The index of each monotone solution equals -1. Therefore the degree is different from zero for any number of solutions if at least one such solution exists.

The model of competition of species

We consider the system of equations

W 1 + W 1 (1 -W 1 -αW 2 ) = 0, W 2 + ρW 2 (1 -βW 1 -W 2 ) = 0, (52)
describing the competition of species in population dynamics. The unknowns are two population densities and the nonlinear expressions represent the rates of reproduction of the two species. All parameters are positive constants and we will suppose that α > 1 and β > 1.

Then the system has four stationary points: P 0 = (0, 0), P 1 = (1, 0), P 2 = (0, 1) and P 3 = ( W1 , W2 ) where 0 < Wi < 1 satisfy W1 + α W2 = 1 and β W1 + W2 = 1. The points P 1 and P 2 are stable while the points P 0 and P 3 are unstable.

The above system is not monotone. However it is possible to obtain a monotone system by introducing the new variable w 1 = W 1 and w 2 = 1 -W 2 . The system of equations now reads:

w 1 + F 1 (w 1 , w 2 ) = 0, w 2 + F 2 (w 1 , w 2 ) = 0, (53)
For these new nonlinearities the equations F 1 (w 1 , w 2 ) = 0 and F 2 (w 1 , w 2 ) = 0 are respectively equivalent to

w 2 = f 1 (w 1 ) and w 1 = f 2 (w 2 ),
where

f 1 (w 1 ) = -w 1 (1 -α -w 1 ) α( + w 1 ) , f 2 (w 2 ) = (1 -w 2 + )w 2 β(1 -w 2 ) .
For all > 0 sufficiently small the functions f 1 and f 2 are monotone functions of their respective arguments (Figure 2), and the system of equations

w 2 = f 1 (w 1 ), w 1 = f 2 (w 2 ),
possesses three solutions. Let us denote the two exterior points by w ± and the intermediate point by w . We have w + = (0, 0) and w -→ (1, 1) as → 0. For the perturbed nonlinearities the pulse is the solution of the system:

w 1 + F 1 (w 1 , w 2 ) = 0, w 2 + F 2 (w 1 , w 2 ) = 0, w i (±∞) = 0, i = 1, 2, (62)
while the traveling wave problem reads:

u 1 + cu 1 + F 1 (u 1 , u 2 ) = 0, u 2 + cu 2 + F 2 (u 1 , u 2 ) = 0, u(±∞) = w ± . (63)
The existence of travelling waves for the perturbed system is investigated in [START_REF] Volpert | Elliptic partial differential equations[END_REF] where the following result is derived. Let us note that the limiting value c 0 in Proposition 4.1 is the speed of the travelling wave for the system (59)-(60). In particular if c 0 > 0, then c > 0 for all positive and sufficiently small . This remark enables us to apply Theorem 1.1 to derive the existence of pulses for the perturbed problem for sufficiently small . Proposition 4.2. Assume that c 0 > 0. Then, for all > 0 sufficiently small, Problem (62) possesses a solution w such that w (0) = 0, w (x) < 0 for x > 0, w (∞) = 0.

(64)

Pulses for the original problem

The existence of some pulse for Problem (53) is related to the sign of the speed c 0 as stated in the following result.

Theorem 4.3. System (53) has a solution satisfying

w (0) = 0, w (x) < 0 for x > 0, w(∞) = 0
if and only if the speed c 0 in problem ( 59)-( 60) is positive.

Proof. Again we refer to [START_REF] Marion | Existence of pulses for a monotone reaction-diffusion system[END_REF] for the proof of the non-existence part in Theorem 4.3 for c 0 ≤ 0.

Hereafter we assume c 0 > 0 and we aim to derive the existence of a pulse. This will proved by taking the limit → 0 for the solutions w provided by Proposition 4.2.

Since 0 < w (x) < w -and w -→ (1, 1) as → 0, the functions w are bounded in L ∞ (R) independently of sufficiently small. Hence classical regularity estimates yield that the Hölder norm in C 2+α (R + ) of w is also bounded independently of sufficiently small.

Consequently there exists a subsequence still denoted by w locally convergent to some limiting function w. The function w satisfies the equations (53), w(x) ≥ 0 and w (x) ≤ 0 for x ≥ 0, w (0) = 0.

Let us first check that w(0) = (0, 0).

Arguing by contradiction suppose that w (0) converges to the origin as → 0.

Recalling the definition (61) it is easy to check the existence of some neighborhood of the origin in R 2 + such that for all sufficiently small and for each point w > 0 in this neighborhood at least one component of F (w) is negative. Due to its convergence property, w (0) enters this neighborhood and at least one component of F (w (0)) is negative. Then due to the equations the second derivative at x = 0 of the corresponding component of w is positive, and this function cannot be decreasing. Similar arguments provide w(0) = (0, 1), w(0) = (1, 0).

Next let us verify that

w(0) > (0, 0). ( 65
)
If this is not true then either w 1 (0) = 0 and w 2 (0) > 0 or w 1 (0) > 0 and w 2 (0) = 0. In the first case w 1 (x) ≡ 0 and w 2 satisfies

w 2 + F 2 (0, w 2 ) = 0, w 2 ≥ 0, w 2 ≤ 0, w 2 (0) = 0.
Since F 2 (0, w 2 ) ≤ 0 this provides readily w 2 = 0 and w 2 ≡ c 2 . The constant c 2 should be such that c 2 > 0 and (0, c 2 ) is a zero of F . The only possibility is c 2 = 1 but this is impossible since we noted that w(0) = (0, 1). In the second case similar arguments provide ( w 1 , w 2 ) ≡ (1 -α, 0) which is also impossible since this point is not a zero of F This shows (65).

The above arguments yield the existence of some constant η > 0 such that for sufficiently small the two components of w satisfy:

w ,1 (0) > η, w ,2 (0) > η (66)
(indeed otherwise there exists a sequence w converging to some w and at least one component of w(0) vanishes).

In order to verify the convergence of w to the origin as x → ∞, it is sufficient to prove that the functions w are bounded independently of > 0 sufficiently small in the (weighted) norm of E 1 µ . Since they are uniformly bounded in the Hölder norm without weight, it is sufficient to prove that the norm sup x |w (x)µ(x)| is uniformly bounded. Let us recall that solutions w converge to zero exponentially at infinity. So this norm is bounded for each solution. Suppose that these functions are not uniformly bounded with respect to in the weighted norm. Then there is a sequence of solutions w k (with 0 < w k (x) < w k -) for which

sup x≥0 |w k (x)µ(x)| → ∞ as k → ∞.
Let δ > 0 be sufficiently small so that the exponential decay of the solutions gives the estimate

|w k (x)µ(x)| ≤ M,
for those values of x such that |w k (x)| ≤ δ, with some constant M > 0 independent of k. Choosing δ < η where η is given by (66), we can select x k so that |w k (x k )| = δ and

|w k (x)µ(x)| ≤ M for x ≥ x k . (67)
If the values x k are uniformly bounded, then the values |w k (x)µ(x)| are uniformly bounded for 0 ≤ x ≤ x k since w k (x) ≤ w k . Together with (67), this provides the required estimate for all x ≥ 0. Suppose that x k → ∞. Consider the sequence of functions u k (x) = w k (x+ x k ). We can choose a subsequence converging to some limiting function u in C 2 loc (R). Then u is a non increasing function defined on the whole axis and satisfies the equation u + F ( u) = 0.

The bound (67) yields that u(∞) = (0, 0) while u(-∞) = u -, where u -is such that F (u -) = 0 and u -= (0, 0) since | u(0)| = δ. Hence we obtain a solution of system (59) with c = 0. Let us show that this is not possible. Indeed, if u -is the unstable zero of F in (w + , w -), then a solution of system (21) exists only for negative c [START_REF] Volpert | Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations[END_REF], [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]. Let us note, that there are two unstable zeros, w 1 = 0, w 2 = 1 and w 1 = (1 -α)/(1 -αβ), w 2 = βw 1 . If u -= w -(stable point), then by virtue of the assumption of Theorem 4.3, c > 0. Hence the function u can not exist, and the sequence x k is bounded.

There remains to show that w (x) < 0 for all x > 0 (component-wise). Indeed, suppose that w i (x 0 ) = 0 for some i and for some x 0 > 0. Assuming for example i = 1 we differentiate the first equation of system (53). Setting

v = -w 1 it provides : -v -a(x) v = b(x),
where

a(x) = ∂F 1 ∂w 1 ( w) , b(x) = - ∂F 1 ∂w 2 ( w) w 2 (x).
Since b(x) ≥ 0, v(x) ≥ 0 and v(x 0 ) = 0, the positiveness theorem yields v(x) ≡ 0 hence w 1 (x) ≡ 0. As already noted this implies w 2 ≡ 0 hence w 2 ≡ 0 which is impossible. The proof of Theorem 4.3 is completed.

Remark. Systems (53) and ( 52) are equivalent thanks to some appropriate change of variable. Hence Theorem 4.3 provides the existence of some even solution W of system (52) satisfying

W 1 (±∞) = 0, W 2 (±∞) = 1, W 1 (x) > 0 and W 2 (x) < 1 for x ∈ R, (68)
if and only if c 0 > 0. Similarly, if c 0 < 0, we obtain the existence of some even solution of (52) such that:

W 1 (±∞) = 1, W 2 (±∞) = 0, W 1 (x) < 1 and W 2 (x) > 0 for x ∈ R. ( 69
)
Indeed let us compare these two cases. In the first one, after the change of variables w 1 = W 1 , w 2 = 1 -W 2 we considered the solution of (53) with zero limits at infinity. It is proved that it exists if and only if the wave in (59) with the limits u i (-∞) = 1, u i (∞) = 0 has a positive speed. In terms of the original variables, these limits are given by (see (58)):

U 1 (-∞) = 1, U 1 (∞) = 0, U 2 (-∞) = 0, U 2 (∞) = 1. ( 70
)
If we consider a solution of (52) with limits (69), then the change of variables is different and reads: w1 = 1 -W 1 , w2 = W 2 . A study similar to the one for the first change of variable can be carried out. In particuler, the wi have required to have zero limits at infinity. From the analogous theorem, we conclude that such solution exists if and only if the corresponding wave with the limits Applying Lemma 4.4 to the function G we see that the speed of the wave solution of u + cu + G(u) = 0, u(±∞) = w ± , equals zero. Hereafter we denote by ũ some solution of ũ + G(ũ) = 0, ũ(±∞) = w ± (74) (the solution is unique up to space translation). Next since β > α we have F 1 (u) = G 1 (u) and F 2 (u) > G 2 (u) for 0 < u i < 1.

(75)

Hence we can apply Theorem 3.2 to the functions F and G and we conclude that c 0 ≥ 0. We now aim to show that c 0 > 0. Arguing by contradiction let us suppose that c 0 = 0. Consider the Cauchy problem

∂v ∂t = ∂ 2 v ∂x 2 + F (v), (76)
with the initial condition v(x, 0) = ũ(x), where ũ is a solution of problem (74). Since ũ + F (ũ) = F (ũ) -G(ũ) ≥ 0 (the inequality is strict for the second component of the vector), ũ is a lower function and v(x, t) increases as a function of t for each x fixed. By virtue of the global asymptotic stability of travelling waves for monotone systems, v(x, t) converges to a solution u(x) of problem (57)-(58). Hence

u(x) > ũ(x), ∀x ∈ R. (77)
For any h ∈ R the function ũ(x -h) is also a solution of problem (74). If h is sufficiently large, then the inequality u(x) > ũ(x -h) (78)

does not hold for some x. Let h 0 be the supremum of the h for which (78) holds for all x ∈ R. If u(x) ≥ ũ(x -h 0 ) for all x and u(x 0 ) = ũ(x 0 -h 0 ) for some x 0 , then we obtain a contradiction with the positiveness theorem. We will now consider the case where such finite x 0 does not exist and shows that this case also leads to some contradiction. According to the assumptions above, there a sequence h n such that h n > h 0 , h n → h 0 as n → ∞ and u(x n ) = ũ(x n -h n ) for some x n and for one of the components of the solutions. The sequence x n can not be bounded. Consider the case where x n → ∞.

Let us take some x * such that F (u(x * )) < 0 and some x n > x * . Such x * exists since u(x) ∼ p exp(-λx) where p > 0 is the eigenvector of the matrix F (0) corresponding to the negative eigenvalue, λ is a positive number. We consider the boundary value problem for the equation (76) on the halfaxis x > x * with the boundary condition v(x * , t) = u(x * ). It has a unique stationary solution u(x). Moreover, this solution is globally asymptotically stable. Indeed, the initial condition v(x, 0) = u(x * ) provides an upper solution and the initial condition v(x, 0) = 0 a lower solution. Both of them converge to the stationary solution u(x) by virtue of its uniqueness.

On the other hand, the function max(u(x), ũ(x -h n )) is a lower function. Therefore the corresponding solution is greater than u(x) and it cannot converge to u(x) since it increases in time. This contradiction proves that x n cannot tend to infinity. Similarly, it can be proved that it cannot go to -∞. This contradiction proves the theorem.

Fig. 1

 1 Fig. 1 Zero lines of the functions G i and the supports of the functions φ i (left). More detailed structure of the support of the functions φ i (right). Inner rectangles show where these functions equal 1.

Fig. 2

 2 Fig. 2 Left: zeros of the functions F 1 (bold solid) and F 2 (bold dashed). Right: zeros of the functions F 1 (bold solid) and F 2 (bold dashed).

Proposition 4 . 1 .

 41 For all > 0 sufficiently small Problem (63) possesses a solution for a unique value of c denoted by c . Moreover, c is a continuous function of .

Acknowledgements

The second author was partially supported by the Ministry of Education and Science of the Russian Federation (the agreement number 02.a03.21.0008) and by the program PICS CNRS 6583 Matbio.

where

These functions satisfy the conditions

if w 1 > 0 and w 2 < 1.

We will also need to consider the system of equations

describing travelling wave solutions of the system of competition of species. We will look for its solutions with the limits at infinity

The equivalent problem in terms of the variables

with the limits :

Existence and uniqueness of travelling wave, that is of solutions of problem (57)-(58) (or equivalently (59)-( 60)) is known (note that the inequality (56) is not strict on the closed unit square, but the result still holds, see [START_REF] Volpert | Travelling wave solutions of parabolic systems[END_REF]).

Hereafter we will denote by c 0 the corresponding wave speed. The wave speed c 0 can be positive, zero or negative depending on the values of parameters. If it is positive the wave propagates in the direction of the values (U 1 , U 2 ) = (1, 0) reached at -∞. Some elements on the sign of the wave speed can be found in Section 4.2.

Existence of pulses for Problem (53)-(56)

Pulses for a perturbed system

As already mentioned the nonlinearity in (53) is such that the system is monotone. However the conditions on the zeros lines stated in ( 5)-( 8) are not satisfied (see Figure 2 below). This leads us to introduce the following modified functions involving the parameter > 0:

(61) ũi (-∞) = 1, ũi (∞) = 0, i = 1, 2 has a positive speed. In terms of the original variables, these limits read:

The wave with such limits has a positive speed if and only if the speed of the wave with the limits (70) is negative. Indeed, it is sufficient to replace U i (x) by U i (-x), i = 1, 2. Thus, depending on the sign of the speed, either there is a solution with limits (68) or a solution with limits (69) (and no solutions if c 0 = 0).

The sign of the wave speed

We proved the existence of pulses when the wave speed c 0 in problem ( 57)-(58) (or equivalently ( 59)-( 60)) is positive. Here we aim to present some simple conditions on the parameters ρ, α and β which guarantee that c 0 > 0.

We start with a case where c 0 = 0 that will be useful afterwards.

Lemma 4.4. Suppose ρ = 1 and α = β. Then the speed c 0 of the wave solution of (57)-( 58) is equal to zero.

Proof. Let us argue by contradiction and suppose c 0 > 0 (the case c 0 < 0 is similar). The functions z 1 (x) = U 2 (-x) and z 2 (x) = U 1 (-x) satisfy the problem

where c * = -c 0 < 0. Since the problem (72)-(73) coincides with the problem (57)-(58), then we conclude that the latter has two different waves, one with a positive speed and another one with a negative speed. We obtain a contradiction with the uniqueness of the wave Our main result is the following one. Proof. For ρ, α and β satisfying the above conditions let us set

and