Existence Results for Nonlinear Anisotropic Elliptic Equation

In this work, we shall be concerned with the existence of weak solutions of anisotropic elliptic operators \(Au +\sum_{i=1}^{N}g_{i}(x, u, \nabla u)+\sum_{i=1}^{N}H_{i}(x, \nabla u)=f-\sum_{i=1}^{N} \frac{\partial }{\partial x_{i}}k_{i}\) where the right hand side \(f\) belongs to \(L^{p^{'}_{\infty}}(\Omega)\) and \(k_{i}\) belongs to \(L^{p_{i}^{'}}(\Omega)\) for \(i=1,...,N\) and \(A\) is a Leray-Lions operator. The critical growth condition on \(g_{i}\) is the respect to \(\nabla u\) and no growth condition with respect to u, while the function \(H_{i}\) grows as \(|\nabla u|^{p_{i}-1}\).

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1. Introduction

In this paper we study the existence of weak solutions to anisotropic elliptic equations with homogeneous Dirichlet boundary conditions of the type

where is a bounded open subset of RN (N 2) with Lipschitz continuous boundary. The operator Au = PNi =1 @ @xi ai (x;u;ru) is a Leray-Lions operator such that the functions ai , gi and Hi are the Carathodory functions satisfying the following conditions for all

where ; ;bi are some positive constants, for i = 1; :::;N and L : R+ ! R+is a continuous and non decreasing function. The right hand side f and ki for i = 1; :::;N are functions belonging to Lp0 1and Lp0 i = pi pi?1 ;p0 1 = p1 p1?1 with p1 = maxfp;p+g where p+ = maxfp1; :::;pNg, p = 1 1N PNi =11 pi and p = Np N?p . Since the growth and the coercivity conditions of each ai for all i = 1; :::;N depend on pi , we have need to use the anisotropic Sobolev space. We mention some papers on anisotropic Sobolev spaces (see e.g.[1]-[5]). If pi = p for all i = 1; :::;N, we refer some works such as by Guib´e in [6], by Monetti and Randazzo in [7] and by Y. Akdim, A. Benkirane and M. El Moumni in [8]. In [3], L.Boccardo, T. Gallouet and P. Marcellini have studied the problem (1) when ai (x;u;ru) = @u @xi pi?1 @u @xi, gi = 0, Hi = 0, ki = 0 and f = is Radon’s measure. In [5], F. Li has proved the existence and regularity of weak solutions of the problem (1) with gi = 0, Hi = 0, ki = 0 for all i = 1; :::;N and f belongs to Lm with m > 1. In [9], R. Di Nardo and F. Feo have proved the existence of weak solution of the problem (1) when gi = 0 for all i = 1; :::;N. In [10], we have proved the existence and uniqueness of weak solution of the problem (1) but when Au = ? PNi =1 @ @xi ai (x;ru) (ai depending only on x and ru). In this work, we prove the existence of weak solutions of the problem (1), based on techniques related to that of Di castro in [11] and to the recent work’s Di Nardo and F. Feo in [9].

2. Preliminaries

Let be a bounded open subset of RN (N 2) with Lipschitz continuous boundary and let 1 < p1; :::;pN < 1 be N real numbers, p+ = maxfp1; :::;pNg;p? = minfp1; :::;pNg and ?!p
= (p1; :::;pN). The anisotropic Sobolev space (see [12])

3. Assumptions and Definition

We consider the following class of nonlinear anisotropic elliptic homogenous Dirichlet problems

4. Main results

In this section we prove the existence of at least a weak solution of the problem (1). We consider the approximate problems.

4.1. Approximate problems and a prior estimates

Let

Proof: Let A be a positive real number, that will be chosen later, Referring to lemma 1. Let us fix s 2 f1; :::; tg and let us use Tk(us) as test function in problem 13, using (5), (8), Young’s and H¨older’s inequalities and proposition 1 we obtain

here and in what follows the constants depend on the data but not on the function u. Using condition (10), H¨older’s and Young’s inequalities, lemma 1 and proposition 1 we get

4.2. Strong convergence of Tk(un)

4.3. Existence results

Theorem 1 Assume that p < N and (5)-(12) hold. Thenthere exists at least a weak solution of the problem (1).

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Existence Results for Nonlinear Anisotropic Elliptic Equation