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Solutions of semiclassical states for perturbed p-Laplacian equation with critical exponent
Boundary Value Problems volume 2014, Article number: 243 (2014)
Abstract
In this paper, we study semiclassical states for perturbed p-Laplacian equations. Under some given conditions and minimax methods, we show that this problem has at least one positive solution provided that ; for any , it has m pairs of solutions if , where ℰ, are sufficiently small positive numbers. Moreover, these solutions in as .
1 Introduction and main results
In this paper, we consider the existence and multiplicity of semiclassical solutions of the following perturbed p-Laplacian equation:
where , is the p-Laplacian operator with , , is the Sobolev critical exponent, is a nonnegative potential, is bounded positive coefficient, and is a p-superlinear but subcritical function.
Such types of equations have been derived as models of several physical phenomena and have been the subject of extensive study in recent years. For example, solutions to (1.1) for , are related to the solitary wave solutions for quasilinear Schrödinger equations,
where , is a given potential, κ, ħ are real constants and ρ, are real functions. The quasilinear equation (1.2) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of . In the case , (1.2) models the superfluid film equation in fluid mechanics by Kurihara [1]. In the case , (1.2) models the self-channeling of a high-power ultra short laser in matter (see [2]–[5]). For more physical motivations and more references dealing with applications, we can refer to [6]–[10] and references therein.
Taking in (1.2), E is some real constant. It is clear that solves (1.2) if and only if solves the following elliptic equation:
with , and .
When , the semilinear problem has been studied extensively under various hypotheses on the potential and the nonlinearities. See, for example, [11]–[24] and the references therein.
When , , , , we can refer to [9], [25]–[29], and so on. Here positive or sign-changing solutions were obtained by using a constrained minimization argument, or a Nehari method, or a technique of changing variables. We remark that among the above three methods, the last one, which was first proposed in [28], is most effective for the power nonlinearity case since this argument can transform the quasilinear problem to a semilinear one and an Orlicz space framework was used as the working space.
It is worth pointing out that the critical exponent case was mentioned as an open problem in [29], where the authors observed that the number 22∗ behaves like a critical exponent for (1.3). In [30], for , the authors treated the case where the nonlinearity has critical exponential growth, that is, h behaves like as . For , when satisfies radially symmetrical, periodic, and some geometric conditions, Moameni [31] obtained the existence of nonnegative solutions for (1.3) with the critical growth case; when satisfied asymptotic and periodic condition. In [24], [32], the authors prove the existence of ground state solutions for (1.3) with or . In the present paper, we will consider a class of quasilinear Schrödinger equations with a nonperiodic potential function in , . In fact, we will investigate the existence of solutions for the critical growth case when the parameter ε goes to zero, i.e., the semiclassical problems for the critical quasilinear Schrödinger equation (1.1). It is well known that in this case the laws of quantum mechanics must reduce to those of classical mechanics, and it describes the transition between quantum mechanics and classical mechanics. As far as we know, there are few papers considering the existence and concentration of semiclassical states for quasilinear Schrödinger equations. For instance, in [33], [34], using a suitable Trudinger-Moser inequality in and a penalization technique, the authors established the existence of semiclassical solutions for the critical exponent case via the mountain pass lemma.
However, it seems that there is almost no work on the existence of semiclassical solutions to the quasilinear problem on involving critical nonlinearities and generalized potential . Fortunately, Ding and Lin [35] have been concerned with the existence and multiplicity of semiclassical solutions of the following perturbed nonperiodic quasilinear Schrödinger equation:
Later, Yang and Ding [36] extended (1.4) to the following quasilinear Schrödinger equation:
Inspired by [36], we will extend the existence and multiplicity of solutions for (1.5) to the general case for (1.5) with , . Moreover, the corresponding problem becomes more complicated: first, is not a Hilbert space when ; secondly, the weak continuity of operator in is difficulty to establish.
In this paper, we make the following assumptions:
(V1): and there is such that the set has finite Lebesgue measure.
(V2):.
(K):, .
(h1):, , uniformly in x as .
(h2): There are and such that
(h3): There are , , such that and .
A typical example satisfying (h1)-(h3) is the function with and being positive and bounded.
Our main results of this paper are as follows.
Theorem 1.1
Let (V1)-(V2), (K), and (h1)-(h3) hold. Then for anythere issuch that ifthen problem (1.1) has at least one positive solutionsatisfying
-
(i)
and
-
(ii)
Moreover, inas.
Theorem 1.2
Assume that (V1)-(V2), (K), and (h1)-(h3) hold, and. Then for anyandthere issuch that if, problem (1.1) has at least m pairs of solutions, , , which satisfy the estimates (i) and (ii) in Theorem 1.1. Moreover, inas.
These results are new for the p-Laplacian equation and are a generalization of the results in [36].
Our goal is to prove the existence of semiclassical solutions of (1.1) by a variational approach. A function is called a weak solution of (1.1) if and for all we have
where . We point out that we cannot apply directly a variational method here because of the natural functional corresponding to (1.1) given by
Because the nonhomogeneous term prevents us from working directly with the functional , which is not well defined in since, for , may hold. The other difficulty is the lack of compactness due to the unboundedness of the domain and the appearance of the Sobolev critical exponent . To overcome these difficulties we generalize an argument developed by Liu et al. in [28] for , (see also [37]). We make the change of variables , and reformulate the problem into a new one which has an associated functional that is well defined and is of class on .
Before we end this section, some notations are in order. We use to denote the integral , denotes the usual norm . In the whole paper, C denotes a generic constant, which may vary from line to line.
The rest of this paper is organized as follows: in Section 2, we describe the analytic setting where we restate the problems in equivalent form by replacing with other than the usual scaling (see [38]), due to the non-autonomy of nonlinearities. In Section 3, we show that the corresponding energy functional satisfies the (PS) condition at the levels less than with some independent of λ. Thus in Section 4 we construct minimax levels less than for all λ large enough. We prove our main results in Section 5.
2 Equivalent variational problems
Let , then (1.1) reads
for . And we introduce the space
which is a Banach space with norm
By (V1), we know that the embedding is continuous. Note the norm is equivalent to the norm defined by
for each . It is clear that, for each , there exists (independent of λ) such that if
Let S be the best Sobolev constant,
We observe that the natural variational functional for (2.1)
is not still well defined in the general function space E. To overcome this difficulty we generalize an argument developed by Liu et al. in [28] for , (see also [37] for ). We make the change of variables , where f is defined by
Thus we collect some properties of f.
Lemma 2.1
The functionenjoys the following properties:
-
(1)
f is uniquely defined function and invertible.
-
(2)
for all .
-
(3)
for all .
-
(4)
as .
-
(5)
for all .
-
(6)
for all .
-
(7)
as .
-
(8)
There exists a positive constant C such that
-
(9)
.
Proof
Similar to [37]. To prove (1), it is sufficient to remark that the function
has a bound derivative. The point (2) is immediate by the definition of f. Inequality (3) is a consequence of (2) and the fact that is an odd and concave function for . Next, we prove (4). As a consequence of the mean value theorem for integrals, we see that
Since , we get
To show item (5), we integrate and we obtain
Using the change of variables , we get
thus (5) is proved for . For , we use the fact is odd. The first inequality in (6) is equivalent to . To show the inequality, we study the function , defined by . Since and using the definition of f, we obtain, for all ,
and the first inequality in (6) is proved. The second inequality in (6) is obtained in a similar way.
Now by point (4) it follows that and the inequality (6) implies that for all
Thus is a nondecreasing function for and this together with estimate (5) shows item (7). Point (8) is an immediate consequence of (4) and (7). Point (9) is obtained from the definition of f. □
After the change of variables, can be reduced to the following functional:
which is on the usual Sobolev space . Moreover, the critical points of are the weak solutions of the following equation:
Now we can restate Theorem 1.1 and Theorem 1.2 as follows.
Theorem 2.2
Let (V1)-(V2), (K), and (h1)-(h3) hold. Then for anythere issuch that ifthen problem (2.3) has at least one positive solutionsatisfying
-
(i)
and
-
(ii)
Moreover, inas.
Theorem 2.3
Let (V1)-(V2), (K), and (h1)-(h3) hold, and. Then for anyandthere issuch that if, problem (2.3) has at least m pairs of solutions, , , which satisfy the estimates (i) and (ii) in Theorem 2.2. Moreover, inas.
Remark 2.4
To prove the existence of positive solutions, we may consider in E
where , then and critical points of are positive solutions for (2.3).
3 Behaviors of (PS) sequences
Let E be a real Banach space and be a function of class . We say that is a (PS) c sequence if and . is said to satisfy the (PS) c condition if any (PS) c sequence contains a convergent subsequence.
The main result of the section is the following compactness result.
Lemma 3.1
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Letbe a (PS) c sequence for. Thenandis bounded in E.
Proof
Let be a (PS) c sequence for , we have
where as .
By (h3) and Lemma 2.1(6), we deduce
Hence combining (3.1) and (3.2), for n large enough,
which implies that there exists such that
Taking the limit in (3.2), we can obtain .
In the following, we need to show is bounded in E. From (3.3), we need to prove that is bounded.
By (V2),
and using Lemma 2.1(8),
These estimates imply that is bounded in E. □
From Lemma 3.1, we know that every (PS) c sequence is bounded, hence, without loss of generality, we may assume in E and , in for , and a.e. for . Obviously, v is a critical point of .
Lemma 3.2
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Letandbe a bounded (PS) c sequence. Then there is a subsequencesuch that, for each, there exists
for all, where.
Proof
For . Noting that in as , we have, for each ,
and there exists such that
Without loss of generality, we can assume . In particular, for , we deduce
Observe that there exists an such that , and the following relation is satisfied:
We have
From Lemma 2.1(5), we know
for all .
For , we only need Lemma 2.1. □
Remark 3.3
From the proof of Lemma 3.2, we can find the same subsequence such that the result of Lemma 3.2 holds for both and .
Let be a smooth function satisfying if , if . Define . Clearly,
Lemma 3.4
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Letbe defined as in Lemma 3.2, then we have
uniformly inwith.
Proof
From (3.5) and local compactness of the Sobolev embedding, for any ,
uniformly in .
Let . By (2.2)
and, for any , it follows from (3.4) that
for all . By (h1), (h2), and Lemma 2.1(2), (5), and (6), we have, for all ,
Therefore, using Lemma 3.2 and Remark 3.3,
which implies the conclusion as required. □
Lemma 3.5
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Letbe the defined in Lemma 3.2, then we have, as,
-
(i)
;
-
(ii)
.
Proof
By (h1)-(h3) and Lemma 2.1, similar to the proof of Lemma 3.4, it is not difficult to check that
By (3.5) and the Brezis-Lieb lemma, we can deduce that
Recalling that, for any fixed , there exists such that, for all ,
therefore,
Using Lemma 2.1(3), we obtain
Applying the Lebesgue dominated convergence theorem, we know that as . Since is bounded and
we deduce that
Similarly, we can obtain
These, together with the facts and as , give conclusion (i).
To verify conclusion (ii), observe that, for any ,
By (3.5) and Lemma 3.2 in [39], we can check that
Hence we have
By Lemma 2.1(6) and (5), we have
Then by the Rellich imbedding theorem and the continuity of the Nemytskii operator, we obtain
uniformly in . Moreover, since is bounded, using the same arguments as in Lemma 3.4 and (3.6), we obtain
and
uniformly in , proving (ii). □
Lemma 3.6
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Then there exists a constantindependent of λ such that, for any (PS) c sequenceforwith, eitherfor a subsequence or
Proof
Taking
then , by (3.5), if and only if . Assume that has no convergent subsequence. Then . By Lemma 3.5, one also has a subsequence that and .
Denote
where b is the positive constant from assumption of (V1). Since the has a finite measure and in , we see that
From (h1)-(h2), we deduce for any fixed that there exists such that
thus by (K), we can find a constant such that
From Lemma 2.1(5) and (6), (3.7), and (3.8), we know
We have
where . It is easy to see that
From (3.9) and (3.10), we obtain
or, equivalently,
where
The proof is complete. □
From Lemma 3.6, we have the following conclusions.
Lemma 3.7
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Thensatisfies the (PS) c condition for all.
Lemma 3.8
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Thensatisfies the (PS) c condition for all.
4 The mountain pass geometry
Lemma 4.1
Let E be a real Banach space andbe a functional of class of. Assume thatis a closed subset of E which disconnects (arcwise) E into distinct connected componentsand. Suppose further thatand
-
(i)
and there exists such that ;
-
(ii)
there exists such that .
Then J possesses a (PS) c sequence withgiven by
where.
From now on, we consider , and the following lemma implies that possesses the mountain pass geometry.
Lemma 4.2
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. For each λ there is a closed subsetof E which disconnects (arcwise) E into distinct connected componentsand. Thensatisfies:
-
(i)
and there exists such that .
-
(ii)
For any finite-dimensional subspace ,
-
(iii)
For any there exists such that, for each , there is such that and
Proof
-
(i)
First note that, for each λ, . Now, for every , define
Since is continuous, then is a closed subset which disconnects the space E. From (h1)-(h2), for any , there exists such that
From Lemma 2.1(3), we know , and since the embedding from E to , , is continuous, we have
Taking such that , using the Hölder inequality and the Sobolev embedding theorem, we obtain
Furthermore, since is bounded, by Lemma 2.1(5) and the Sobolev embedding theorem, we get
for every . Since , we conclude that there are and such that .
-
(ii)
Observe that, by (h3), . Define the functional by
Then
For any finite-dimensional subspace , we only need to prove
In fact, by Lemma 2.1(8), we get
Thus
Since all norms in a finite-dimensional space are equivalent and , one easily obtains the desired conclusion.
-
(iii)
From Lemma 4.1 and Lemma 4.2(i)-(ii), if satisfies the (PS) c condition for all , then Theorem 2.2 follows from a variant mountain pass theorem. However, in general we do not know if satisfies the (PS) c condition. By Lemma 3.7 for λ large and small enough, satisfies the (PS) condition. Thus we will find a special finite-dimensional subspace by which we construct sufficiently small minimax levels for when λ is large enough.
Recall that
For any , we can choose with and such that . Set
then . Remark that, for ,
where is defined by
It is easy to show that
Since and , there is such that
Thus
Therefore, for all ,
Choosing such that
and taking , from (ii), we can choose large enough and define ; then we get
□
Remark 4.3
For any , one can choose nonnegative such that the function defined by (4.5) is nonnegative. In fact, if is a sequence in with and , then by Kato’s inequality, the absolute value sequence with and , where denotes the set of all continuous functions in with compact supports. Therefore, Lemma 4.2 is still true with the function .
As a consequence of Lemma 4.2 and Remark 4.3, we have the following conclusions.
Corollary 4.4
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. For anythere existssuch that, for each, there isand a (PS)sequencesatisfying
where.
Corollary 4.5
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. For anythere existssuch that, for each, there isand a (PS)sequencesatisfying
where.
5 Proof of the main results
In section, we prove the existence and multiplicity results.
Proof of Theorem 2.2
In virtue of Corollary 4.4, for any , there exists , there is and a (PS) sequence satisfying
where . Lemma 3.7 implies that satisfies the (PS) condition, thus there is such that and , then is a positive solution of (2.1). Moreover, it is well known that a mountain pass solution is a state solution of (2.1).
Since is a critical point of , for ,
where μ is the constant in (h3). Taking yields
and taking gives
Then
which means in as . The proof is completed. □
Remark 5.1
By the same arguments as applied to , we can obtain the existence of positive solutions for (2.3).
In order to obtain the multiplicity of critical points, we will apply the index theory defined by the Krasnoselski genus. Denote the set of all symmetric (in the sense that ) and closed subsets of E by Σ. For each , let be the Krasnoselski genus and
where Σ is the set of all odd homeomorphisms and is the closed symmetric set
such that . Then i is a version of Benci’s pseudoindex [40]. Let
If is finite and satisfies the (PS) condition, then we know all are critical values for .
Proof of Theorem 2.2
Consider the functional , from (h1)-(h3), we know, for each λ, there is a closed subset of E and such that .
In the same way as we have done in Lemma 4.2, for any and , we can choose m functions such that if , and . Let be such that , . Set
and
Then . Observe that, for each ,
and as before
Set
and choose such that
Thus it is easily to obtain
for all . Choose such that
Thus, for any and , there exists such that , we can choose an m-dimensional subspace with .
Since and , we deduce
where defined by (5.1).
It follows from Lemma 3.7, satisfies the (PS) c condition if . Then all are critical values and has at least m pairs of nontrivial critical points satisfying
Therefore, (2.3) has at least m pairs of solutions and must solve problem (2.1). □
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Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions on improving this paper. This research is supported by NSFC: Tianyuan Foundation (11326145, 11326139), and also supported by Hubei Provincial Department of Education (Q20122504) and Youth Science Foundation program of Jiangxi Provincial (20142BAB211010).
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Wang, J., Wang, L. & Zhang, D. Solutions of semiclassical states for perturbed p-Laplacian equation with critical exponent. Bound Value Probl 2014, 243 (2014). https://doi.org/10.1186/s13661-014-0243-y
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DOI: https://doi.org/10.1186/s13661-014-0243-y