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

Ghanmi Abdeljabbar Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia Correspondence should be addressed to Ghanmi Abdeljabbar; abdeljabbar.ghanmi@lamsin.rnu.tn Received 30 September 2013; Accepted 9 December 2013; Published 12 February 2014 Academic Editors: E. Colorado, L. Gasinski, and D. D. Hai Copyright © 2014 Ghanmi Abdeljabbar.This is an open access article distributed under theCreativeCommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper shows the existence and multiplicity of nontrivial solutions of the p-Laplacian problem −Δ pu = (1/σ)(∂F(x, u)/∂u) +

Inspired by the work of Brown and Zhang [10], Nyamouradi [11] treated the following problem: where  is positively homogeneous of degree  * − 1.

ISRN Mathematical Analysis
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: We remark that using assumption (H 1 ), for all  ∈ Ω,  ∈ R, we have the so-called Euler identity: Our main result is the following.
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.

Some Notations and Preliminaries
Throughout this paper, we denote by   the best Sobolev constant for the operators  1, 0 (Ω) →   (Ω), given by where 1 <  ≤  * .In particular, we have with the standard norm Problem ( 1) is posed in the framework of the Sobolev space  =  1, 0 (Ω).Moreover, a function  in  is said to be a weak solution of problem (1) if Thus, by (6) the corresponding energy functional of problem (1) is defined in  by In order to verify   ∈  1 (, R), we need the following lemmas.
Lemma 4 (See Proposition 1 in [13]).Suppose that (, )/ ∈ (Ω × R, R) verifies condition (12).Then, the functional   belongs to  1 (, R), and where ⟨⋅, ⋅⟩ denotes the usual duality between  and  * :=  −1,  (Ω) (the dual space of the sobolev space ).As the energy functional   is not bounded below in , it is useful to consider the functional on the Nehari manifold: Thus,  ∈   if and only if Note that   contains every nonzero solution of problem (1).
Moreover, one has the following result.
Lemma 5.The energy functional   is coercive and bounded below on   .
Proof.If  ∈   , then by (16) and condition (A) we obtain So, it follows from (8) that Thus,   is coercive and bounded below on   .Define Then, by ( 16) it is easy to see that for  ∈   , Now, we split   into three parts Lemma 6. Assume that  0 is a local minimizer for   on   and that  0 ∉  0  .Then,    ( 0 ) = 0 in  −1 (the dual space of the Sobolev space E).Proof.Our proof is the same as that in Brown-Zhang [10, Theorem 2.3].
From now on, we denote by  0 the constant defined by then we have the following.

Proposition 12. (i)
There exist minimizing sequences { +  } in  +  such that (ii) There exist minimizing sequences Proof.The proof is almost the same as that in Wu [14, Proposition 9] and is omitted here.

Proof of Our Result
Throughout this section, the norm   is denoted by ‖ ⋅ ‖  for 1 ≤  ≤ ∞ and the parameter  satisfies 0 < || <  0 .
Theorem 13.If 0 < || <  0 , then, problem (1) has a positive solution  + 0 in  +  such that Proof.By Proposition 12(i), there exists a minimizing sequence { +  } for   on  +  such that Then by Lemma 5, there exists a subsequence {  } and  + 0 in  such that Next, we will show that By Lemma 3, we have where  = /( − 1).