Existence of Positive Weak Solutions for a New Class of Laplacian Nonlinear Elliptic System with Sign-Changing Weights

Abstract

In this paper, by using subsuper solutions method, we study the existence of weak positive solutions for a new class of Laplacian nonlinear elliptic system in bounded domains, when , , and are sign-changing functions that maybe negative near the boundary, without assuming sign conditions on , and .

This presented work is in memory of second author’s father (1910–1999) Mr. Mahmoud ben Mouha Boulaaras

1. Introduction

The study of differential equations and variational problems with nonstandard growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1–15]). Many existence results have been obtained on this kind of problems, see, for example [10, 12, 16–18] and [19–30]. In [31, 32], Fan et al. studied the regularity of solutions for differential equations with nonstandard –-growth conditions.

In this article, we consider the following system:where , is a bounded domain with smooth boundary , , and , and are nonnegative parameters.

In fact, we study the existence of positive solutions to system (1) with sign-changing weight functions , , , and . Due to these weight functions, the extensions are challenging and nontrivial.

These problems arise in some physical models and are interesting in applications at combustion, mathematical biology, and chemical reactions. Our approach is based on the method of sub- and supersolutions (see [6, 23–25, 27, 30]).

We make the following assumptions: , , , are nondecreasing functions such that   

Let be the first eigenvalue of with Dirichlet boundary conditions and be the corresponding eigenfunction with in and for . Let be such that on and on for .

Here, we assume that the weights , , , and take negative values in but require , , , and to be strictly positive in . To be precise, we assume that there exist positive constants , , , , , and such that

Also, let be such that , , , , and

For , we definewhere and ; also assume

2. Existence Result

We give the following two definitions before we give our main result.

Definition 1. Let , be said a weak solution of (1) if it satisfiesfor all .

Definition 2. A pair of nonnegative functions in is called a weak subsolution and supersolution of (1) if they satisfy on :for all .
We shall establish the following result.

Theorem 1. Assume that the conditions hold, , , , and are in and . LetThen, problem (1) has a positive weak solution for each .

Proof of Theorem 1. Let and be such that .
We shall verify thatis a subsolution of (1). Let the test function with . We haveSimilarly,For all with .
Now, on we haveOn the contrary, on we haveand similarly,Therefore, is subsolution of problem (1).
Next, we construct a supersolution of (1). Let be a unique positive solution offor . We denotewhere , , and is a large number to be chosen later. We shall verify that is a supersolution of (1) such that . By , we can choose large enough so thatHence,Using (17),NextBy , choose large so thatThen, from (19), we haveAccording to (19) and (20), we can conclude that is a supersolution of (1). Furthermore, and for large, Thus, there exists a solution of (1) with and . This completes the Proof of Theorem 1.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Authors’ Contributions

All authors contributed equally to this work. They have all read and approved the final manuscript.