#### Abstract

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:(H_{1}) is a function such that
(H_{2}), , and for all .We remark that using assumption (H_{1}), for all , , we have the so-called Euler identity:
Our main result is the following.

Theorem 1. *Under the assumptions (A), (H _{1}), and (H_{2}), 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 Problem (1) is posed in the framework of the Sobolev space . 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 , we need the following lemmas.

Lemma 2. *Assume that is positively homogeneous of degree ; then is positively homogeneous of degree .*

*Proof. *The proof is the same as that in Chu and Tang [4].

In addition, by Lemma 2, we get the existence of positive constant such that

Lemma 3 (see [12], Theorem A.2). *Let and such that
**
Then for every , one has ; moreover the operator defined by is continuous.*

Lemma 4 (See Proposition 1 in [13]). *Suppose that verifies condition (12). Then, the functional belongs to , and
**
where denotes the usual duality between and (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 is a local minimizer for on and that . Then, in (the dual space of the Sobolev space E).*

*Proof. *Our proof is the same as that in Brown-Zhang [10, Theorem 2.3].

Lemma 7. *One has the following:*(i)*if , then ;*(ii)*if , then and ;*(iii)*if , then .*

*Proof. *The proof is immediate from (21), (22), and (23).

From now on, we denote by the constant defined by
then we have the following.

Lemma 8. *If , then .*

*Proof. *Suppose otherwise, that such that . Then for , we have
From the Hölder inequality, (6) and (8), it follows that
Hence, it follows from (27) that
then,
On the other hand, from condition (A), (8) and (26) we have
So,
Combining (30) and (32), we obtain , which is a contradiction.

By Lemma 8, for , we write and define
Then, we have the following.

Lemma 9. *If , then
**
for some depending on , and .*

*Proof. *Let . Then, from (23) we have
So
Thus, from the definition of and , we can deduce that .

Now, let . Then, using (6) and (8) we obtain
this implies that
In addition, by (18) and (38)
Thus, since , we conclude that for some . This completes the proof.

For with , set Then, the following lemma holds.

Lemma 10. *For each with , one has the following:*(i)*if , then there exists unique such that and
*(ii)*if , then there are unique such that and
*

*Proof. *We fix with and we let
Then, it is easy to check that achieves its maximum at . Moreover,

(i) We suppose that . Since as , for and for . There is a unique such that .

Now, it follows from (14) and (27) that
Hence, . On the other hand, it is easy to see that for all
Thus, .

(ii) We suppose that . Then, by (A), (8) and the fact that we obtain
Then, there are unique and such that , , and . We have , and
Thus,
This completes the proof.

For each with , set Then we have the following.

Lemma 11. *For each with , one has the following:*(i)*if , then there exists a unique such that and
*(ii)*if , then there are unique such that and
*

*Proof. *For with , we can take
and similar to the argument in Lemma 9, we obtain the results of Lemma 10.

Proposition 12. *
(i) There exist minimizing sequences in such that
**
(ii) There exist minimizing sequences in such that
*

*Proof. *The proof is almost the same as that in Wu [14, Proposition 9] and is omitted here.

#### 3. Proof of Our Result

Throughout this section, the norm is denoted by for and the parameter satisfies .

Theorem 13. *If , then, problem (1) has a positive solution 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 in such that
This implies that as .

Next, we will show that
By Lemma 3, we have
where . On the other hand, it follows from the Hölder inequality that
Hence, as .

By (57) and (58) it is easy to prove that is a weak solution of (1).

Since
then by (57) and Lemma 9, we have as . Letting , we obtain
Now, we aim to prove that strongly in and .

Using the fact that and by Fatou's lemma, we get
This implies that
Let ; then by Brézis-Lieb Lemma [3] we obtain
Therefore, strongly in .

Moreover, we have . In fact, if then, there exist such that and . In particular we have . Since
there exists such that . By Lemma 10, we have
which is a contradiction.

Finally, by (63) we may assume that is a nontrivial nonnegative solution of problem (1).

Theorem 14. *If , then, problem (1) has a positive solution in such that
*

*Proof. *By Proposition 12(ii), there exists a minimizing sequence for on such that
Moreover, by (23) we obtain
So, by (38) and (72) there exists a positive constant such that
This implies that
By (70) and (71), we obtain clearly that is a weak solution of (1).

Now, we aim to prove that strongly in . Supposing otherwise, then
By Lemma 9, there is a unique such that . Since , for all , we have
which is a contradiction. Hence strongly in .

This imply that
By Lemma 5 and (74) we may assume that is a nontrivial solution of problem (1).

Now, we begin to show the proof of Theorem 1: by Theorem 13, we obtain that for all , problem (1) has a nontrivial solution . On the other hand, from Theorem 14, we get the second solution . Since , then and are distinct.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.