Existence of Nontrivial Solutions of p-Laplacian Equation with Sign-Changing Weight Functions

This paper shows the existence and multiplicity of nontrivial solutions of the p-Laplacian problem for with zero Dirichlet boundary conditions, where is a bounded open set in , if , if ), , is a smooth function which may change sign in , and . The method is based on Nehari results on three submanifolds of the space . 1. Introduction In this paper, we are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation: where is a bounded domain of , if , if , , is positively homogeneous of degree ; that is, holds for all and the sign-changing weight function satisfies the following condition: (A) with ,？？ , and？？ . In recent years, several authors have used the Nehari manifold and fibering maps (i.e., maps of the form , where is the Euler function associated with the equation) to solve semilinear and quasilinear problems. For instance, we cite papers [1–9] and references therein. More precisely, Brown and Zhang [10] studied the following subcritical semilinear elliptic equation with sign-changing weight function: where . Also, the authors in [10] by the same arguments considered the following semilinear elliptic problem: where . Exploiting the relationship between the Nehari manifold and fibering maps, they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter crosses the bifurcation value. Inspired by the work of Brown and Zhang [10], Nyamouradi [11] treated the following problem: where is positively homogeneous of degree . In this work, motivated by the above works, we give a very simple variational method to prove the existence of at least two nontrivial solutions of problem (1). In fact, we use the decomposition of the Nehari manifold as vary to prove our main result. Before stating our main result, we need the following assumptions:(H1) is a function such that (H2) , , and for all .We remark that using assumption (H1), for all , , we have the so-called Euler identity: Our main result is the following. Theorem 1. Under the assumptions (A), (H1), and (H2), there exists such that for all , problem (1) has at least two nontrivial nonnegative solutions. This paper is organized as follows. In Section 2, we give some notations and preliminaries and we present some technical lemmas which are crucial in the proof of Theorem 1. Theorem 1 is proved in Section 3. 2. Some Notations and Preliminaries Throughout this paper, we denote by the best Sobolev constant for the operators , given by where . In particular, we have with the standard norm