Research

# Infinitely many solutions for a class of quasilinear elliptic equations with p-Laplacian in R N

Gao Jia*, Jie Chen and Long-jie Zhang

Author Affiliations

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

For all author emails, please log on.

Boundary Value Problems 2013, 2013:179  doi:10.1186/1687-2770-2013-179

 Received: 15 December 2012 Accepted: 22 July 2013 Published: 6 August 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we study the multiplicity of solutions for a class of quasilinear elliptic equations with p-Laplacian in . In this case, the functional J is not differentiable. Hence, it is difficult to work under the classical framework of the critical point theory. To overcome this difficulty, we use a nonsmooth critical point theory, which provides the existence of critical points for nondifferentiable functionals.

MSC: 35J20, 35J92, 58E05.

##### Keywords:
quasilinear elliptic equations; nondifferentiable functional; p-Laplacian; multiple solutions

### 1 Introduction and main results

Recently, the multiplicity of solutions for the quasilinear elliptic equations has been studied extensively, and many fruitful results have been obtained. For example, in [1], Shibo Liu considered the existence of multiple nonzero solutions of the Dirichlet boundary value problem

(1.1)

where , denotes the p-Laplacian operator, Ω is a bounded domain in with smooth boundary Ω.

Moreover, Aouaoui studied the following quasilinear elliptic equation in [2]:

(1.2)

and proved the multiplicity of solutions of the problem (1.2) by using the nonsmooth critical point theory. One can refer to [3,4] and [5] for more results.

In this paper, we shall investigate the existence of infinitely many solutions of the following problem

(1.3)

where , and , is a given continuous function satisfying

In order to determine weak solutions of (1.3) in a suitable functional space E, we look for critical points of the functional defined by

(1.4)

where . Under reasonable assumptions, the functional J is continuous, but not even locally Lipschitz. However, one can see from [4,6] and [7] that the Gâteaux-derivative of J exists in the smooth directions, i.e., it is possible to evaluate

for all and .

Definition 1.1 A critical point u of the functional J is defined as a function such that , , i.e.,

(1.5)

Our approach to study (1.3) is based on the nonsmooth critical point theory developed in [8] and [9]. Dealing with this class of problems, the main difficulty is that the associated functional is not differentiable in all directions.

The main goal here is to establish multiplicity of results for (1.3), when is odd and is even in s. Such solutions for (1.3) will follow from a version of the symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [10,11]. Compared with problem (1.2) in [2], problem (1.3) is much more difficult, since the discreteness of the spectrum is not guaranteed. Therefore, we only consider the first eigenvalue .

To state and prove our main result, we consider the following assumptions.

Suppose that and .

(H1) Let be a function such that

• for each , is measurable with respect to x;

• for a.e. , is a function of class with respect to s;

• there exist such that

(1.6)

(1.7)

(H2) There exist , and such that

(1.8)

(H3) Let a Carathéodory function satisfy , a.e. and

(1.9)

where θ is the same as that in (H2).

(H4) There exists such that

(1.10)

where is a positive constant.

Example 1.1 Let . The following function satisfies hypotheses (H1) and (H2)

and the corresponding constants are

Example 1.2 The following function satisfies hypotheses (H3) and (H4)

On the other hand, we define the operator . It follows from [12] that the discreteness of the spectrum is not guaranteed. Hence, we only consider the first eigenvalue , where

Next, we can state the main theorem of the paper.

Theorem 1.1Assume thatandsatisfy (H1)-(H4). Moreover, letand, a.e. , . If there exists a positive numberμsuch that, then problem (1.3) has infinitely many distinct solutions in, i.e., there exists a sequence, satisfying (1.3) and, as.

To explain our result, we introduce some functional spaces. We define the reflexive Banach space E of all functions with the norm .

Such a weighted Sobolev space has been used in many previous papers, see [13] and [14]. Now, we give an important property of the space E, which will play an essential role in proving our main results.

Remark 1.1 One can easily deduce and for . More details can be found in [2].

Throughout this paper, let denote the norm of E and () means that converges strongly (weakly) in corresponding spaces. ↪ stands for a continuous map, and ↪↪ means a compact embedding map. C denotes any universal positive constant unless specified.

The paper is organized as follows. In Section 2, we introduce the nonsmooth critical framework and preliminaries to our work. In Section 3, we give some lemmas to prove the main result. Finally, the proof of Theorem 1.1 is presented in Section 4.

### 2 Nonsmooth critical framework and preliminaries

Our results are based on the techniques of nonsmooth critical point theory. In this section, we recall some basic tools from [8] and [9].

Definition 2.1 Let be a metric space, let be a continuous functional and . We denote by the supremum of the σ’s in such that there exist and a continuous map , satisfying

The extended real number is called the weak slope of I at u.

Note that the notion above was independently introduced in [15], as well.

Definition 2.2 Let be a metric space, let be a continuous functional and . We say that I satisfies , i.e., the Palais-Smale condition at level c, if every sequence in X with and admits a strongly convergent subsequence.

In order to treat the Palais-Smale condition, we need to introduce an auxiliary notion.

Definition 2.3 Let c be a real number. We say that functional I satisfies the concrete Palais-Smale condition at level c ( for short) if every sequence satisfying

possesses a strongly convergent subsequence in E, where is some real number converging to zero.

Remark 2.1 Under assumptions (H1)-(H4), if the functional J satisfies (1.4), then J is continuous, and for every we have

where denotes the weak slope of J at u.

Remark 2.2 Let c be a real number. If J satisfies , then J satisfies .

Proof Let be a sequence such that

Note that for ,

By Remark 2.1, we have . Taking , the conclusion follows. □

### 3 Basic lemmas

To derive our main theorem, we need the following lemmas. The first lemma is the version of the Ambrosetti-Rabinowitz mountain pass lemma [10,11] and [16].

Lemma 3.1LetXbe an infinite-dimensional Banach space, and letbe a continuous even functional satisfyingfor every. Assume that

(i) there exist, and a subspaceof finite codimension such that

(ii) for every finite-dimensional subspace, there existssuch that

Then there exists a sequenceof critical values ofIwith.

Lemma 3.2Ifis a critical point ofJ, then.

Proof For , , consider the real functions , and defined in ℝ by

(3.1)

and . Denoting and , we can take as a test function in (1.5). Therefore,

Noting that and , we get

From (1.10) and the fact we deduce

Since a.e. in and in E as . It follows from that

Denote . If , then the result is true. In the following discussion, is assumed. By (1.6), we obtain

(3.2)

Note that , then we can get

(3.3)

On the other hand, we have

which implies that

(3.4)

Eventually, one can deduce from (3.2)-(3.4) that

(3.5)

By Theorem 5.2 of [17], we get that . Replacing by , we can similarly prove that . We conclude that , and the proof of Lemma 3.2 is completed. □

Lemma 3.3Letbe a bounded sequence inEwith

(3.6)

whereis a sequence of real numbers converging to zero. Then there existssuch thata.e. inand, up to a subsequence, is weakly convergent touinE. Moreover, we have

(3.7)

i.e., uis a critical point ofJ.

Proof Since is bounded in E, and there is a (see [18]) such that, up to a subsequence,

Moreover, since satisfies (3.6), by Theorem 2.1 of [19], we have, up to a further subsequence, a.e. in .

We will use the device of [20]. We consider the test functions

(3.8)

where , and . According to (1.6) and (1.7), we have

Since (3.6) holds by density for every , we can put in (3.6) and obtain that

(3.9)

On the other hand, note that

(3.10)

One can deduce from (3.10) and Fatou’s lemma that

(3.11)

We consider the test functions with , and , , ,

This together with (3.11) can prove that

(3.12)

In a similar way, by considering the test functions , it is possible to prove that

(3.13)

From (3.12) and (3.13), it follows that

(3.14)

Finally, we can deduce (3.7) from (3.14). □

Remark 3.1 (see [21])

Let be a sequence in E satisfying (3.6). Then and

(3.15)

In the following lemma, we will prove the boundedness of a sequence under (1.6), (1.8) and (1.9).

Lemma 3.4Letandbe a sequence inEsatisfying (3.6) and

(3.16)

Thenis bounded inE.

Proof Calculating , from (3.15) and (3.16), we obtain

From (1.8) and (1.9), it follows that

(3.17)

Moreover, there exist and such that

Therefore, denoting , we obtain from (3.17) that

(3.18)

By virtue of hypothesis (H3), we know that there exist and such that

(3.19)

From (3.18) and (3.19), it follows that

(3.20)

On the other hand, by Hölder’s inequality and Young’s inequality, for all , there exists such that

(3.21)

Using (3.20) and (3.21), we get

(3.22)

Choosing in (3.22), we find that is bounded in E. □

Lemma 3.5Letbe the same as that in Lemma 3.3. Then, up to a subsequence, converges strongly touinE.

Proof By Lemma 3.3, we know that u is a critical point of the functional J. Then, from Lemma 3.2, we get . Therefore, taking as a test function in (3.7), we get

(3.23)

By virtue of is bounded in E, we can assume that there exists satisfying

By Lemma 3.3, a.e. in . Then by Fatou’s lemma, we have

(3.24)

Moreover, by and , we get

(3.25)

(3.26)

By using (3.23)-(3.26) and passing to limit in (3.15), we obtain

(3.27)

On the other hand, by Lebesgue’s dominated convergence theorem and the weak convergence of to u in E, we get

(3.28)

(3.29)

(3.30)

Moreover, since and are bounded in , then we have

Therefore, from the definition of weak convergence, we obtain

(3.31)

(3.32)

Combining (3.27)-(3.32), it follows that

It is well known that the following inequality

(3.33)

holds for any , and . Therefore,

According to (1.6), we conclude that converges strongly to u in E. □

Lemma 3.6For every real numberc, the functionalJsatisfies.

Proof Let be a sequence in E satisfying (3.6) and (3.16). By Lemma 3.4, is bounded in E. Therefore, the conclusion can be deduced from Lemma 3.5. □

### 4 Proof of Theorem 1.1

It is easy to check that the functional J is continuous and even. Moreover, by Remark 2.2 and Lemma 3.6, J satisfies for every .

On the other hand, from (1.4), (1.6), (1.9) and (1.10), for , we have

(4.1)

We discuss (4.1) in the following two cases:

In case , we get

In case , by the definition of , we get

i.e., . Therefore, if λ satisfies , there exist small enough and such that

Hence, condition (i) of Lemma 3.1 holds with .

Now we consider a finite-dimensional subspace W of E. Let and . From (1.6), we have

(4.2)

By virtue of (1.9) and (1.10), we know that there exist , satisfying a.e. and a positive constant such that

(4.3)

Combining (4.2)-(4.3), we have

(4.4)

Since W is finite-dimensional, then all norms of W are equivalent. From (4.4), there exists such that

In view of , we deduce that the set is bounded in E and condition (ii) of Lemma 3.1 holds. By Lemma 3.1, the conclusion follows.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.

### Acknowledgements

The authors express their sincere thanks to the referees for their valuable criticism of the manuscript and for helpful suggestions. This work has been supported by the Natural Science Foundation of China (No. 11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

### References

1. Liu, SB: Multiplicity results for coercive p-Laplacian equations. J. Math. Anal. Appl.. 316, 229–236 (2006). Publisher Full Text

2. Aouaoui, S: Multiplicity of solutions for quasilinear elliptic equations in . J. Math. Anal. Appl.. 370(2), 639–648 (2010). Publisher Full Text

3. Alves, CO, Carrião, PC, Miyagaki, OH: Existence and multiplicity results for a class of resonant quasilinear elliptic problems on . Nonlinear Anal.. 39, 99–110 (2000). Publisher Full Text

4. Canino, A: Multiplicity of solutions for quasilinear elliptic equations. Topol. Methods Nonlinear Anal.. 6, 357–370 (1995)

5. Squassina, M: Existence of multiple solutions for quasilinear diagonal elliptic systems. Electron. J. Differ. Equ.. 1999, 1–12 (1999)

6. Arcoya, D, Boccardo, L: Critical points for multiple integrals of the calculus of variations. Arch. Ration. Mech. Anal.. 134, 249–274 (1996). Publisher Full Text

7. Arcoya, D, Boccardo, L: Some remarks on critical point theory for nondifferentiable functionals. NoDEA Nonlinear Differ. Equ. Appl.. 6, 79–100 (1999). Publisher Full Text

8. Corvellec, JN, Degiovanni, M, Marzocchi, M: Deformation properties of continuous functionals and critical point theory. Topol. Methods Nonlinear Anal.. 1, 151–171 (1993)

9. Degiovanni, M, Marzocchi, M: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl.. 167(4), 73–100 (1994)

10. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text

11. Silva, EAB: Critical point theorems and applications to differential equations. Ph.D. thesis, University of Wisconsin-Madison (1988)

12. Brasco, L, Franzina, G: On the Hong-Krahn-Szego inequality for the p-Laplace operator. Manuscr. Math.. 141, 537–557 (2013). Publisher Full Text

13. Bartsh, T, Wang, ZQ: Existence and multiplicity results for some superlinear elliptic problems on . Commun. Partial Differ. Equ.. 20, 1725–1741 (1995). Publisher Full Text

14. Rabinowitz, PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text

15. Katriel, G: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 11, 189–209 (1994)

16. Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)

17. Ladyzenskaya, OA, Uralceva, NN: Equations aux dérivées partielles de type elliptiques, Dunod, Paris (1968)

18. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc.. 88, 486–490 (1983)

19. Boccardo, L, Murat, F: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal.. 19, 581–597 (1992). Publisher Full Text

20. Boccardo, L, Murat, F, Puel, JP: Existence de solutions non bornées pour certaines équations quasi-linéaires. Port. Math.. 41, 507–534 (1982)

21. Brezis, H, Browder, FE: Sur une propriété des espaces de Sobolev. C. R. Math. Acad. Sci. Paris. 287, 113–115 (1978)