Weak homoclinic solutions of anisotropic discrete nonlinear system with variable exponent

We prove the existence of weak solutions for an anisotropic homoclinic discrete nonlinear system. Suitable Hilbert spaces and norms are constructed. The proof of the main result is based on a minimization method. We also extend the problem by using generalized penality and source functions.


Introduction
In this paper, we investigate the existence of weak solutions for the following anisotropic nonlinear discrete system. where ∆u i (k) = u i (k + ) − u i (k) is the forward di erence operator and α, a, f i are functions to be de ned later. The di erence equations is the discrete counterpart of PDEs and are usually studied in connection with numerical analysis. In this way, the main operator in problem (1.1) −∆ a(k − , ∆u i (k − )) can be seen as a discrete counterpart of the anisotropic operator − ∂ ∂x j a x, ∂ ∂x j u i .
In the recent years, increasing attention has been paid to the study of di erential and partial di erential equations involving variable exponent conditions (see [2,4,7,13,[15][16][17]19]). The interest in the study of these problems was stimulated by their applications in elastic mechanics, in uid dynamics and calculus of variations. For information on modeling physical phenomena by equations involving the p(x)-growth condition, we refer the reader to ( [10][11][12]) and the references therein. The study of homoclinic connections for boundary value problems has had a certain impulse in recent years, motivated by applications in various biological, physical and chemical models, such has phase-transition, physical processes in which the variable transits from an unstable equilibrium to a stable one, or frontpropagation in reaction-di usion equations. The purpose of this paper is to prove the existence of solutions to the problem (1.1) under appropriate assumptions on α, a and f . We adapt the classical minimization methods used for the study of anisotropic PDEs to prove the existence of solution of problem (1.1). Note that we examine anisotropic di erence system on unbounded discrete interval, typically, on the whole set Z, with asymptotic conditions of homoclinic type. Remark that in the reference [8] the authors studied a particular case, where α ≡ , u(.) ∈ R and the function f does not depend on the solution u. The result we present in this work is more general. Indeed, we consider a system of n equations where the source function f depends on the solution u(.) = u (.), · · · , un(.) ∈ R n . Also we make an extension of the main problem where we observe a competition phenomena between the functions α and σ. Our approach is critical points theorem, namely the idea of the proof is to transfer the problem of the existence of solution for (1.1) into the problem of existence of a minimizer for some associated energy functional. The remaining part of this paper is organized as follows. Section 2 is devoted to mathematical preliminaries. The main existence result is proved in Section 3. In the Section 4, we give an extension of our system.

Mathematical background
In this section, we will introduce some notations, de nitions and preliminary facts which are used throughout this paper.
To construct appropriate function spaces and apply critical point theory in order to investigate the existence of solutions for system (1.1), we introduce the following basic notations and results which will be used in the proofs of our main results. We assume that the function p : Z −→ ( , +∞) and we denoted by For any i ∈ Z[ , n] = { , · · · , n} we introduce the spaces On the space H p(.) i , we introduce the Luxemburg norm Then is a norm on the space H ,p(.) i .
We de ne the space For the data a and f i , for any k ∈ Z, we assume the following.
Z≥ is necessary a nite set and |u i (k)| < ∞ for any k ∈ Z≥ since u i ∈ H ,p(.) i . As Z≥ is a nite set, then Thus lim |k|→+∞ u i (k) = .

Existence of weak solutions
In this section, we state and prove our main results in this paper. Hence, we rst de ne the weak solution of problem (1.1).

De nition 3.1. A weak solution of problem (1.1) is a function u ∈ H such that
Note that, since H is a nite dimensional space, the weak solution coincides with the classical solution of the problem (1.1).

Theorem 3.1. Assume that (H ) − (H ) holds. Then, there exists a weak solution of the problem (1.1).
To prove this we de ne the energy functional J : H −→ R by

Lemma 3.1. The functional J is well de ned on H and is of class C H, R with the derivative given by
for all u, v ∈ H.
Indeed, let and We have by using assumptions (2.1), (2.3) and (2.10) that By using (2.7) and (2.8) We have The energy functional J is well de ned on H.
Using the method applied in [9], Lemma 3.4, it is not di cult to see that the functional I, L derivative are give by On the other hand, for all u, v ∈ H, there exists h i ∈ R n such that The functional J is clearly of class C

Lemma 3.2. The functional J is lower semi-continuous.
The functional Λ is completely continuous and weakly lower semi-continuous. We have to prove the semi-continuity of I. Finally We conclude that J is weakly lower semi-continuous.
To prove the coerciveness of the functional J, we may assume that ||u|| ,p(.) > . We deduce from the above inequality (a) of Proposition 2.1, that Namely J is coercive. Besides, for ||u|| ,p(.) ≤ , using inequality (b) of Proposition 2.1, we have Since J is weakly lower semi-continuous, bounded from below and coercive on H, using the relation between critical points of J and problem (1.1), we deduce that J has a minimizer which is a weak solution of problem (1.1).

An extension
In this section we are going to show that the existence result obtained for System (1.1) can be extended to more general discrete boundary value of the form  and Rodrigue Sanou wrote mathematical formula, bring up the proves and did all the calculus with the other authors. All the authors read and approved the nal manuscript.