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 .
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  studied the following subcritical semilinear elliptic equation with sign-changing weight function: where . Also, the authors in  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.
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 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 .
Lemma 3 (see , 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 ). 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 .
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
On the other hand, from condition (A), (8) and (26) we have
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
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
(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  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.
C.-M. Chu and C.-L. Tang, “Existence and multiplicity of positive solutions for semilinear elliptic systems with Sobolev critical exponents,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5118–5130, 2009.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. H. Rasouli and G. A. Afrouzi, “The Nehari manifold for a class of concave-convex elliptic systems involving the -Laplacian and nonlinear boundary condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 10, pp. 3390–3401, 2010.View at: Publisher Site | Google Scholar | MathSciNet