Abstract
We are concerned with the following elliptic equations with variable exponents: in , where the function is of type with continuous function and satisfies a Carathéodory condition. The purpose of this paper is to show the existence of at least one solution, and under suitable assumptions, infinitely many solutions for the problem above by using mountain pass theorem and fountain theorem.
MSC: 35D30, 35J60, 35J90, 35P30, 46E35.
Similar content being viewed by others
1 Introduction
The differential equations and variational problems with -growth conditions have been much interest in recent years since they can model physical phenomena which arise in the study of elastic mechanics, electro-rheological fluid dynamics and image processing, etc. We refer the readers to [1]–[4] and references therein.
In this paper, we establish some results about the existence and multiplicity of nontrivial weak solution to nonlinear elliptic equations of the -Laplacian type,
where the function is of type with continuous function and satisfies a Carathéodory condition. The essential interest in studying problem (B) starts from the presence of the -Laplace type operator , which is small perturbation of the -Laplace operator . The study for the -Laplacian problems has been extensively considered by several authors in various ways; see for example [5]–[8] and references therein. Fan and Zhang [6] established the existence of solutions for the -Laplacian Dirichlet problems on bounded domains by using the variational method. For the case of the entire domain , the existence and multiplicity results of solutions for the -Laplacian equations have been discussed in [5]. Concerning the -Laplace type operator, Mihăilescu and Rădulescu in [3] investigated a multiplicity result for quasilinear nonhomogeneous problems with Dirichlet boundary conditions by adequate variational methods and a variant of mountain pass theorem which are crucial tools for finding solutions to elliptic problems. In particular, in order to obtain the existence of solutions for equations like (B) in [3], they assume that the functional Φ induced by φ is uniform convex, that is, there exists a constant such that
for all and , where Ω is a bounded domain in . When and , it is well known that this condition is not applicable for the p-Laplacian problems because the function is not uniformly convex for . Recently the authors in [9] have studied the existence of infinitely many solutions for a class of quasilinear elliptic problems involving the -Laplace type operator with nonlinear boundary conditions without using the uniform convexity of Φ.
The aim of this paper is to show the existence of at least one nontrivial solution and infinitely many nontrivial solutions for problem (B) without the assumption of the uniform convexity of the functional Φ as in [9]; see also [10]. We give our main results in a more general setting than those of [5], [6] because (B) is a problem which involves the usual -Laplacian operator. Especially, our proof as regards the existence of infinitely many nontrivial solutions for (B) is different from those of [5], [6], [9].
This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces. In Section 3, under certain conditions on φ and f, we establish several existence results of nontrivial weak solutions for problem (B) by employing as the main tools the variational principle.
2 Preliminaries
In this section, we recall some definitions and basic properties of the variable exponent Lebesgue spaces and the variable exponent Lebesgue-Sobolev spaces which will be treated in the next sections. For a deeper treatment on these spaces, we refer to [11]–[13].
Set
For any , we define
For any , we introduce the variable exponent Lebesgue space
endowed with the Luxemburg norm
The dual space of is , where .
The variable exponent Sobolev space is defined by
where the norm is
It has the following equivalent norm:
Lemma 2.1
([11])
The spaceis a separable, uniformly convex Banach space, and its conjugate space iswhere. For anyand, we have
Lemma 2.2
If, then for any, , and,
Lemma 2.3
([11])
Denote
Then
-
(1)
(=1; <1) if and only if (=1; <1), respectively;
-
(2)
if , then ;
-
(3)
if , then .
Remark 2.4
([5])
Denote
Then
-
(1)
(=1; <1) if and only if (=1; <1), respectively;
-
(2)
if , then ;
-
(3)
if , then .
Lemma 2.5
([13])
Letbe such that, for almost all. Ifwith, then
-
(1)
if , then ;
-
(2)
if , then .
Lemma 2.6
Letbe an open, bounded set with Lipschitz boundary and letwith. Ifwithsatisfies
for all, then we have
and the imbedding is compact if.
Lemma 2.7
([14])
Suppose thatis Lipschitz continuous with. Letand, for almost all. Then there is a continuous embedding.
In what follows, let be Lipschitz continuous with . We denote by the space , and be a dual space of X. Furthermore, denotes the pairing of X and its dual and Euclidean scalar product on , respectively.
3 Existence of solutions
In this section, we shall give the proof of the existence of nontrivial weak solutions for problem (B), by applying the mountain pass theorem, fountain theorem, and the basic properties of the spaces and .
Definition 3.1
We say that is a weak solution of problem (B) if
for all .
We assume that is the continuous derivative with respect to v of the mapping , , that is, . Suppose that φ and satisfy the following assumptions:
(J1) The equalities
hold, for almost all and for all .
(J2) satisfies the following conditions: is measurable, for all , and is continuous, for almost all .
(J3) There are a function and a nonnegative constant b such that
for almost all and for all .
(J4) is strictly convex in , for all .
(J5) The relation
holds, for all and , where d is a positive constant.
Let us define the functional by
The analog of the following lemma can be found in [3]. However, we will give the proof of those because our growth condition is slightly different from that of [3].
Lemma 3.2
Assume that (J1)-(J3) and (J5) hold. Then the functional Φ is well defined on X, and its Fréchet derivative is given by
Proof
A simple calculation as in [3] implies that the functional Φ is well defined on X. For a fixed , it is clear that . Let , then given and , by the classical mean value theorem, there exist with and such that
and
Since
it is easy to obtain
for a positive constant C. Hence . Since implies and , it follows from the Lebesgue dominated convergence theorem that
Let and be an operators defined by
Then the operators and are continuous on X. In fact, for any , let in X as . Then there exist a subsequence and functions v, in for such that as , and and , for all and for almost all . Without loss of generality, we assume that and . Then we have
and
Hence (J3) implies that the integrands at the right-hand sides in the above estimates are dominated by integrable functions. Since the function φ satisfies (J2) and in X as , we obtain and as , for almost all . Therefore, the Lebesgue dominated convergence theorem tells us that in and in as . Thus, and are continuous on X. From the Hölder inequality, we have
for all , and thus
Consequently, the operator is continuous on X. □
Now we will show that the operator is a mapping of type , which plays a key role in obtaining our main results. To do this, we first prove the following useful result.
Lemma 3.3
Assume that (J1)-(J5) hold. If the sequenceinsuch that
as, forand for almost all, theninas.
Proof
Let be a subsequence of the sequence in satisfied
as for any . Then there exists such that, for almost all ,
This together with assumptions (J3), (J5), and Young’s inequality imply that
for almost all , and hence
for almost all , where d is the positive constant from (J5). Since , the sequence is bounded, and then the sequence is bounded in . By passing to a subsequence, we can assume that as , for some . Then we obtain as and the relation (3.2) implies that
Since it follows from assumption (J4) and Proposition 25.10 in [15] that φ is monotone on X, this relation occurs only if , that is, in as . Since these arguments hold for any subsequence of the sequence , we conclude that in as . □
Next we give the following assertion, which is based on the idea of the proof in [16]; see [17] for the case of bounded domain in .
Lemma 3.4
Assume that (J1)-(J5) hold. Then the functionalis convex and weakly lower semicontinuous on X. Moreover, the operatoris a mapping of type, i.e., ifin X asand, thenin X as.
Proof
Let be a sequence in X such that in X as and
It follows from in X as that as . Since Φ is strictly convex by (J4), it is obvious that the operator is monotone, that is,
By (3.3) and (3.4), we have
Hence the sequences and converge to 0 in and as , respectively. By Lemma 3.3, we have in and in ℝ as , for almost all . Then (3.3) holds in the stronger form
It follows from the convexity of Φ that
and hence we obtain by (3.5). Since the functional Φ is strictly convex and -functional on X, it follows that Φ is weakly lower semicontinuous on X. Then it is immediate that . Thus it implies
Consider the sequence in defined pointwise by
From (J1) and (J4), it is clear that the sequence . Since is continuous, for almost all , we obtain as , for almost all . Hence, by the Fatou lemma and (3.6), we have
Thus we get
that is,
Then by assumption (J5), , we conclude that in X as . □
Until now, we considered some properties for the integral operator corresponding to the divergence part in problem (B). To deal with our main results in this section, we need the following assumptions for f. Denoting , we assume that
(H1), , and , for all .
(H2), for some with and , for all .
(F1) satisfies the Carathéodory condition in the sense that is measurable, for all , and is continuous, for almost all .
(F2)f satisfies the following growth condition: For all ,
where q and m are given in (H1) and (H2), respectively.
(F3) There exists a positive constant θ such that and
(F4), as uniformly, for all .
Then it follows from assumption (F2) that
(F2′):, for all .
Define the functional by
Then it is easy to check that and its Fréchet derivative is
for any . Next we define the functional by
Then it follows that the functional and its Fréchet derivative is
for any .
Lemma 3.5
Assume that (H1)-(H2) and (F2) hold. Then Ψ andare weakly strongly continuous on X.
Proof
Proceeding the argument analogous to Lemma 3.2 of [5], it implies that the functionals Ψ and are weakly strongly continuous on X. □
With the aid of Lemma 3.5, we prove that the energy functional satisfies the Palais-Smale condition ((PS)-condition for short). This plays a key role in obtaining the existence of a nontrivial weak solution for the given problem.
Lemma 3.6
Assume that (J1)-(J5), (H1)-(H2), and (F1)-(F3) hold. Thensatisfies the (PS)-condition, for all.
Proof
Note that is of the type , since is weakly strongly continuous. Let be a (PS)-sequence in X, i.e., and as . Since is of type and X is reflexive, it suffices to verify that the sequence is bounded in X. Suppose that , in the subsequence sense. By assumption (J5), we deduce that
where θ is a positive constant from (F3). By condition (F3), we have
For n large enough, we may assume that . Then it follows from (J5) and Remark 2.4(2) that
Since and , this is a contradiction. □
We are now prepared to prove our main results for the existence of at least one solution and infinitely many solutions for problem (B), following the basic idea in [10]. The following consequence can be established by applying the mountain pass theorem with Lemmas 3.4 and 3.6.
Theorem 3.7
Assume that (J1)-(J5), (H1)-(H2) and (F1)-(F4) hold. Then problem (B) has a nontrivial weak solution, for all.
Proof
Note that . Since satisfies the (PS)-condition, it is enough to show the geometric conditions in the mountain pass theorem, i.e.,
-
(1)
there is a positive constant R such that
-
(2)
there exists an element v in X satisfying
Let us prove the condition (1). By Lemma 2.7, there exists a positive constant such that . Let be small enough such that for the positive constant d from (J5). By assumptions (F2) and (F4), for any , there exists a positive constant denoted by such that
for all . Assume that . Then it follows from (J5) and Lemmas 2.1, 2.5(2), 2.7, and Remark 2.4 that
for a positive constant . Then it follows that
Since , there exist small enough and such that when .
Next we show the condition (2). Meanwhile, observe that (J4) implies that, for all , , and ,
Indeed, let us define . Then we have
It implies that
Integrating this inequality over , we have
and so
Hence we find that (3.8) holds. In a similar way, we find that condition (F3) implies
for all , , and .
Take . Then it follows from (3.8) and (3.9) that
where . Since , we see that as . Therefore satisfies the geometry of the mountain pass theorem. □
Now, adding the oddity on f and using the fountain theorem in Theorem 3.6 in [18], we shall demonstrate infinitely many pairs of weak solutions for problem (B). To employ the fountain theorem, we consider the following situation. This lemma holds for a reflexive and separable Banach space.
Lemma 3.8
([6])
Let W be a reflexive and separable Banach space. Then there areandsuch that
and
Let us denote , , and .
Theorem 3.9
Assume that (J1)-(J5), (H1)-(H2), and (F1)-(F4) hold. Ifholds, for all, thenhas a sequence of critical pointsin X such thatas.
Proof
Obviously, is an even functional and satisfies (PS)-condition. It is enough to show that there exist such that
-
(1)
as ;
-
(2)
,
for k large enough.
Denote
Then as . In fact, suppose that it is false. Then there exist and the sequence in such that
for all . Since the sequence is bounded in X, there exists such that in X as and
for . Hence we get . But we have
which provides a contradiction.
For any , it follows from (F2′), (J5) and Lemmas 2.1 and 2.7, and Remark 2.4 that
where and is a positive constant. Choose . Then as since and as . Hence, if and , we deduce that
for a positive constant , which implies (1).
To show (2), from (F4) we see that, for , there exists such that
for all and for all . Then we know that there exists such that , for almost all , and
for almost all and all . In fact, by (F3), we have, for all ,
Then it follows that
and thus
Hence we get
Similarly, we obtain
for all . Thus, , for almost all and all , where . Also assumptions (F2) and (F3) imply that and .
Assume that . For any , by (J5), (F3), (3.11), (3.12), Lemmas 2.1, 2.3, 2.7, and Remark 2.4, we have
By Hölder’s inequality and Lemma 2.7, we deduce that
for a positive constant . Notice that in the finite dimensional subspace , the norm is equivalent to the norm . Therefore, it follows from (3.13) that
for a positive constant . Since , we obtain
and thus we can choose . This completes the proof. □
The following consequence is the other way to show the existence of infinitely many pairs of weak solutions for the given problem (B) without assumption (F4).
Theorem 3.10
Assume that (J1)-(J5), (H1)-(H2), and (F1)-(F3) hold. In addition, suppose that there existandwith, for almost all, such that
for almost alland for allwhere. Ifholds, for all, thenhas a sequence of critical pointsin X such thatas.
Proof
Obviously, is an even functional and satisfies (PS)-condition. It is enough to show that there exist such that
-
(1)
as ;
-
(2)
,
for k large enough. The same argument in Theorem 3.9 implies (1).
To show (2), for , write
It is easy to see that . For with and , by (3.14) we have
for a positive constant . Since , it follows from (3.15) that
and thus we can choose . This completes the proof. □
References
Diening L, Harjulehto P, Hästö P, Ru̇žička M: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin; 2011.
Edmunds DE, Rákosník J:Density of smooth functions in .Proc. R. Soc. Lond. Ser. A 1992, 437: 229-236. 10.1098/rspa.1992.0059
Mihăilescu M, Rădulescu V: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 2006, 462: 2625-2641. 10.1098/rspa.2005.1633
Ru̇žička M: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin; 2000.
Fan X, Han X:Existence and multiplicity of solutions for -Laplacian equations in . Nonlinear Anal. 2004, 59: 173-188.
Fan X, Zhang QH:Existence of solutions for -Laplacian Dirichlet problem. Nonlinear Anal. 2003, 52: 1843-1852. 10.1016/S0362-546X(02)00150-5
Kim, IH, Kim, Y-H: Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents. Manuscr. Math. (in press)
Mihăilescu M, Rădulescu V: On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent. Proc. Am. Math. Soc. 2007, 135: 2929-2937. 10.1090/S0002-9939-07-08815-6
Boureanu M-M, Preda F: Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 2012, 19: 235-251. 10.1007/s00030-011-0126-1
Lee, JS, Kim, Y-H: Existence and multiplicity of solutions for nonlinear elliptic equations of p-Laplace type in . (submitted)
Fan X, Zhao D:On the spaces and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617
Kim IH, Kim Y-H:Positivity of the infimum eigenvalue for equations of -Laplace type in .Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-214
Szulkin A, Willem M: Eigenvalue problem with indefinite weight. Stud. Math. 1995, 135: 191-201.
Fan X: Sobolev embeddings for unbounded domain with variable exponent having values across N . Math. Inequal. Appl. 2010, 13: 123-134.
Zeidler E: Nonlinear Functional Analysis and Its Applications. II/B. Springer, New York; 1990.
Colasuonno F, Pucci P, Varga C: Multiple solutions for an eigenvalue problem involving p -Laplacian type operators. Nonlinear Anal. 2012, 75: 4496-4512. 10.1016/j.na.2011.09.048
Le VK: On a sub-supersolution method for variational inequalities with leray-lions operators in variable exponent spaces. Nonlinear Anal. 2009, 71: 3305-3321. 10.1016/j.na.2009.01.211
Willem M: Minimax Theorems. Birkhäuser, Basel; 1996.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lee, S.D., Park, K. & Kim, YH. Existence and multiplicity of solutions for equations involving nonhomogeneous operators of -Laplace type in . Bound Value Probl 2014, 261 (2014). https://doi.org/10.1186/s13661-014-0261-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-014-0261-9