Abstract and Applied Analysis

Volume 2014 (2014), Article ID 942092, 14 pages

http://dx.doi.org/10.1155/2014/942092

Research Article

Multiple Solutions for a Class of -Laplacian Equations with Critical Growth and Indefinite Weight

Guoqing Zhang1,2 and Ziyan Yao¹

¹College of Sciences, University of Shanghai for Science and Technology, Shanghai 200093, China

²Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, USA

Received 7 September 2013; Revised 10 December 2013; Accepted 20 December 2013; Published 23 January 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 Guoqing Zhang and Ziyan Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Using the suitable Trudinger-Moser inequality and the Mountain Pass Theorem, we prove the existence of multiple solutions for a class of -Laplacian equations with critical growth and indefinite weight , , , , where , is an indefinite weight, behaves like and does not satisfy the Ambrosetti-Rabinowitz condition, and .

1. Introduction

In this paper, we consider the existence of multiple solutions for the -Laplacian elliptic equations with critical growth and singular potentials

where , , , , , is the -Laplacian, the indefinite weight , and is the classes of rearrangement of ; satisfies the following conditions:

(H1), ,(H2), or , for some ,(H3), where , is the best constant

 that is,

Note that if is a measurable function which satisfies (H2), there exists such that .

Recently, -Laplacian equations had been studied by many authors. Marcos do Ó [1] studied the existence of nontrivial solutions for the following -Laplacian equations with critical growth:

where is bounded smooth domain in . Adimurthi and Sandeep [2] proved that the singular Trudinger-Moser inequality

holds if and only if , where , , , and , and studied the corresponding critical exponent problem. For the unbounded domain, Li and Ruf [3] proved that, if we replace the -norm of in the supermum by the standard Sobolev norm, the supermum can still be finite. Adimurthi and Yang [4] obtained the following Trudinger-Moser inequality

where , , , and , and studied the existence of nontrivial solution for the corresponding -Laplacian equations with critical growth. In particular, using inequality (6) and the Mountain Pass Theorem, Lam and Lu [5] studied the following nonuniformly elliptic equations of -Laplacian type of the form

where , and obtained the existence and multiplicity results of problem (7).

On the other hand, some authors have studied the case for the nonlinear term which does not satisfy the Ambrosetti-Rabinowitz condition. Lam and Lu [6, 7] studied the existence of nontrivial solutions for the -Laplacian equations and systems and polyharmonic equations without Ambrosetti-Rabinowitz conditions, respectively. Miyagaki and Souto [8] discussed a class of superlinear problems for the polynomial case without Ambrosetti-Rabinowitz conditions. Motivated by a suitable Trudinger-Moser inequality, we assume the following growth conditions on the nonlinearity :

(f1)the function is continuous, for some constants and for all ,

(f2), for all , , where

(f3)there exists such that for all , ;(f4), uniformly on . We state our main result in this paper.

Theorem 1. Suppose that (H1)–(H3) and (f1)–(f4) are satisfied and . Furthermore, assume that

(f5), uniformly on , and there exists such that(f6)

uniformly on compact subsets of , where and are defined in Section 3. Then there exists such that, for each , problem (1) has at least two nontrivial weak solutions.

In this paper, as the function is an indefinite weight, we establish a singular Trudinger-Moser inequality (see Lemma 8) and investigate the eigenvalue problem corresponding to problem (1). Using the singular Trudinger-Moser inequality, the eigenvalue problem and the Mountain Pass Theorem, we prove the multiplicity result for problem (1). Furthermore, condition (f2) is used by Lam and Lu [5], and it implies that the function does not satisfy the Ambrosetti-Rabinowitz condition.

The paper is organized as follows. In Section 2, we recall some important lemmas and consider the eigenvalue problem corresponding to problem (1). Section 3 is devote to prove Theorem 1.

2. Preliminary Results

2.1. Key Lemmas

Now, we define the following Sobolev space

and the corresponding norm,

From the Radial Lemma [9, 10], we have

for all being radially symmetric, where is the surface area of the unit sphere in . is a rearrangement of if

where denotes the Lebesgue measure.

Lemma 2 (see [11]). Let satisfy and . Then there exists such that

Proof. Assume that . Since

Then, by (H2), there exists such that

Therefore, we have

Remark 3. In this paper, we denote as positive (possibly different) constants.

Remark 4. If , then satisfies (H1)–(H3).

Lemma 5. If (H1)–(H3) are satisfied, then

(1)the embedding is continuous, for all ;(2)the embedding is compact, for all .

Proof. (1) From Lemma 2 and Sobolev-Poincare inequality, we obtain the conclusion.

(2)Let satisfy for all , and we assume

In view of (H3), for every , there exists such that

Hence, we have

From (19), we have in and is the ball centered at with radius . This together with (21) leads to . Since is arbitrary, we have

Hence, for every , we have

Lemma 6. is a reflexive Banach space.

Proof. Suppose that , , we have

and there exists , using the following inequality

such that

Hence, is uniformly convex. We obtain that is a reflexive Banach space.

Now, we define the functional

then the functional is well defined by Lemma 5. Moreover, is the functional on and ; we have

Hence, the critical point of the functional is the weak solution of problem (1).

Lemma 7. Let , , and ; then for some and , one has

Proof. Let ; is the Schwarz symmetrization of ; we can conclude that

Let . It is easy to obtain that is increasing with respect to . If ; then there holds

Now, we prove that there exists a uniform constant such that, for all radially decreasing symmetric functions and ,

where . In the following, assume that is radially decreasing function in and . Take sufficiently large; that is, . By the radial lemma, for all , we have and

Define the set . Assume that is nonempty; then for all and we have

Since

so we have

Thus, we obtain

From Hardy-Littlewood inequality, we have

Since , we have . Let

Applying the mean value theorem to the function , we obtain that there exists which satisfies

such that

So we have

By , we have

and ,

Obviously, , and

Let ; we obtain

Hence, we obtain that (32) holds. For , , we have

Since and , , that is, , , we have

As the proof of Lemma 7, we can obtain the following.

Lemma 8. For , , and , , and , one has

2.2. The Eigenvalue Problem

We consider the following eigenvalue problem:

Now, we denote the set , and define

where .

Lemma 9 (see [12]). Let , be two continuous functions in differentiable a.e., and

Then , a.e. in if and only if for some .

Proposition 10. Assume that (H1)–(H3) hold; then is the lowest eigenvalue of Problem (50) and is principal.

Proof. From Lemma 2, we have . Furthermore, any minimizing sequence is bounded. Up to a subsequence, there exists such that

Hence, we have

and consequently we have

From Lemma 5, we obtain that is weakly closed in . By the Lagrange Multipliers rule, is an eigenvalue of problem (50). Moreover for any , so that possesses a nonnegative eigenfunction. We conclude that the eigenvalue is principal from Harnack inequality in [13].

Proposition 11. The eigenvalue is isolated. That is, there exists , such that there are no other eigenvalues of problem (50) in the interval .

Proof. Assume by contradiction there exists a sequence of eigenvalue of problem (50) with . Let be an eigenfunction associated with . Then satisfies

We define

The coercivity of the functional implies that is a bounded sequence. Hence is bounded in . So there exists a subsequence (still denoted) and such that

and . On the other hand, we have

and . So we conclude that is an eigenfunction associated with and . Then we conclude from the convergence in measure of the sequence towards that

where denotes the negative set of , which contradicts Proposition 11.

Proposition 12. The first eigenvalue is simple, in the sense that the eigenfunctions associated with it are merely constant multiples of each other.

Proof. Let , be two eigenfunctions associated with . We assume without restriction that , ; then satisfies . Testing it with function , we get

Let , from Lemma 9, we have

The function , where , belongs to and then it is admissible for the weak formulation of , a.e., and

It follows from (62) and (63) that we have

Let ; we have . By Lemma 9, there exists such that .

3. The Proof of Theorem 1

3.1. Palais-Smale Sequence

Now, we check that the functional satisfies the geometric conditions of the Mountain Pass Theorem.

Lemma 13. Suppose that (H1)–(H3) and (f1)–(f5) hold. Then there exists such that, for , there exists such that if . Furthermore, can be chosen such that , as .

Proof. From (f5), for every , there exists such that implies

Moreover, using (f1), for each and , we find a constant such that

Combining (65) and (66), we have

Since the embedding is continuous, we obtain

Thus, we have

Since and and letting , we choose such that . Thus, if is sufficiently small, we find some such that if and even as .

Lemma 14. If , there exists , with , such that .

Proof. Let , with compact support . By (f4), we obtain that for , there exists a positive constant such that for every ,

Then, we have

Choose and and let

we have as . Setting with being sufficient large, we obtain the conclusion.

It is well known that the failure of the (PS) compactness condition creates some difficulties in studying the class of elliptic problems involving critical growth. In Lemma 15, instead of (PS) sequence, we analyze the compactness of Cerami sequences of the functional .

Lemma 15. Let be a Cerami sequence of ; that is,

Then there exists a subsequence of (still denoted by ) and such that

where . Furthermore, is a nontrivial weak solution of problem (1).

Proof. Let , , as ; we have

where as . Let in (76); we have

Suppose that

Set

we have . From [5], we have

However, since is the Cerami sequence at the level , we have that

Then there exists some constant such that

which implies that

Let ; then we have

So we can conclude that

Note that ; by Fatou Lemma, (80), and (85), we get a contradiction. So which means that in .

Let , such that

For any given , let , by (f1); there exists such that

Since , we have

and by (f3), , and , we have

Since in and the embedding is compact from and the Hölder inequality, we have . By Lemma 7, we have .

Let in (89) and ; we get

Note that and ; we suppose that . Since , we have

By (f2) and , we have

Moreover, we have

which is a contraction to (75); this proves that is bounded in . Thus, we have

From Lemma 5, the embedding is compact for all . If , we get

From (f1), the Trudinger-Moser inequality, and the Hölder inequality, we have . From Lemma 2.1 in [14], we have

For any fixed , set

Because is bounded, is a finite set. From Lemma 4.4 in ([4]), for any compact set , we have

Now, we prove that

It is enough to prove that for any and there holds

We take with and on . Then is a bounded sequence. Choosing and in (76), we have

Adapting an argument similar to [4], we have

Since is finite, it follows that a.e. This implies, up to a subsequence that in . Let in (76), and in ; we obtain

Remark 16. The idea and proof of Lemma 15 follow as in Lemma 4.1 in [5].

3.2. Min-Max Value

In order to get a more precise information about the minimax level obtained by the Mountain Pass Theorem, we consider the following sequence of scale which is called the Moser function:

Hence, we have , the support of is the ball , and

Let ; we have

where

From (105), we conclude that , as . Consequently, we have

Lemma 17. Suppose that (H1)–(H3) and (f1)–(f6) hold. Then there exists such that

Proof. Choose as in and such that

where

Suppose, by contradiction, that, for all , we get

where . For each , there exists such that

Thus; we have

From , , we obtain

Let , we have

By (f6), given that , there exist and ; we have

From (116) and (117), for large , we obtain

Let

we have

Hence, the sequence is bounded. Otherwise, up to subsequences, we have , which leads to a contradiction. From (108) and (115) and

it follows that

From [4], we have

Now, using the change of variable

by straight forward computation, we have

which converges to as , where

Finally, let in (118); from (108) and (115), we have

which implies that

Remark 18. The idea and the proof of Lemma 17 come from Lemma 3.6 in [5].

Lemma 19. There exist and with such that for all . In particular, .

Proof. See Lemma 3.3 in [10].

Corollary 20. Under the conditions (H1)–(H3) and (f1)–(f4), if , then one has

From Lemmas 13 and 19, we conclude that

Corollary 21. There exist and with compact support such that, for all ,

Lemma 22. If is a Cerami sequence for at any level with

then possesses a subsequence which converges strongly to a solution of problem (1).

Proof. See Lemma 4.6 in [4].

In conclusion, we have

3.3. Multiplicity Results

In order to prove the existence of the second solution of problem (1) follows by the minimum argument and Ekeland's Variational Principle.

Proposition 23. Under the conditions (H1)–(H3) and (f1)–(f6), there exists such that, for each with , problem (1) has a solution via Mountain Pass Theorem.

Proof. See Proposition 4.1 in [5].

Proposition 24. There exists such that, for each with , (1) has a minimum type solution with , where is defined in (130).

Proof. See Proposition 5.1 in [5].

Proposition 25. If is sufficiently small, then the solutions of problem (1) obtained in Propositions 23 and 24 are distinct.

Proof. By Propositions 23 and 24, there exist sequences , in such that

Suppose by contradiction that . As in the proof of Lemma 15, we have

Hence, by (f2) and (f3) and Generalized Lebesgue's Dominated Convergence Theorem, we obtain that there exists such that

Claim  1. , as . Indeed, by (f2) and (f3), we have

Hence, on the domain and , we have

Since is bounded, and using (137), we have

For , we have

Since is bounded, we have

Combining (139) and (141), we have

Similarly, we also have

Combining (136), (142), and (143), we obtain

Hence, the claim is proved.

Claim  2. . Indeed, we have

From [10], we have

On the other hand, since , we have

By the inequality , we have

Hence, we have in and . It is a contradiction. The proof is complete.

Conflict of Interests

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

Acknowledgments

The authors are grateful to the reviewers for some valuable suggestions. This paper was supported by Shanghai Natural Science Foundation Project (no. 11ZR1424500) and Shanghai Leading Academic Discipline Project (no. XTKX2012).

References

J. Marcos do Ó, “Semilinear Dirichlet problems for the -Laplacian in with nonlinearities in the critical growth range,” Differential and Integral Equations, vol. 9, no. 5, pp. 967–979, 1996.

Adimurthi and K. Sandeep, “A singular Moser-Trudinger embedding and its applications,” Nonlinear Differential Equations and Applications, vol. 13, no. 5-6, pp. 585–603, 2007.

Y. Li and B. Ruf, “A sharp Trudinger-Moser type inequality for unbounded domains in ,” Indiana University Mathematics Journal, vol. 57, no. 1, pp. 451–480, 2008.

Adimurthi and Y. Yang, “An interpolation of Hardy inequality and Trundinger-Moser inequality in and its applications,” International Mathematics Research Notices, vol. 13, pp. 2394–2426, 2010.

N. Lam and G. Lu, “Existence and multiplicity of solutions to equations of -Laplacian type with critical exponential growth in ,” Journal of Functional Analysis, vol. 262, no. 3, pp. 1132–1165, 2012.

N. Lam and G. Lu, “Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 6, pp. 2187–2205, 2012.

N. Lam and G. Lu, “Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition,” Journal of Geometric Analysis, vol. 24, no. 1, pp. 118–143, 2014.

O. H. Miyagaki and M. A. S. Souto, “Superlinear problems without Ambrosetti and Rabinowitz growth condition,” Journal of Differential Equations, vol. 245, no. 12, pp. 3628–3638, 2008.

H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983.

J. Marcos do Ó, E. Medeiros, and U. Severo, “On a quasilinear nonhomogeneous elliptic equation with critical growth in ,” Journal of Differential Equations, vol. 246, no. 4, pp. 1363–1386, 2009.

L. M. Del Pezzo and J. Fernández Bonder, “An optimization problem for the first weighted eigenvalue problem plus a potential,” Proceedings of the American Mathematical Society, vol. 138, no. 10, pp. 3551–3567, 2010.

W. Allegretto and Y. X. Huang, “A Picone's identity for the -Laplacian and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 7, pp. 819–830, 1998.

J. Serrin, “Local behavior of solutions of quasi-linear equations,” Acta Mathematica, vol. 111, pp. 247–302, 1964.

D. G. de Figueiredo, O. H. Miyagaki, and B. Ruf, “Elliptic equations in with nonlinearities in the critical growth range,” Calculus of Variations and Partial Differential Equations, vol. 3, no. 2, pp. 139–153, 1995.