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Multiplicity of solutions for Kirchhoff-type problems involving critical growth
Boundary Value Problems volume 2014, Article number: 210 (2014)
In this paper, by using the concentration-compactness principle and the variational method, we obtain a multiplicity result for Kirchhoff-type problems involving critical growth in bounded domains.
MSC: 35J70, 35B20.
In this paper we deal with the existence and multiplicity of solutions to the following Kirchhoff-type problems involving the critical growth:
where () is an open bounded domain with smooth boundary and λ is a positive parameter. The number is the critical exponent according to the Sobolev embedding.
Much interest has arisen in problems involving critical exponents, starting from the celebrated paper by Brezis and Nirenberg . For example, Li and Zou  obtained infinitely many solutions with odd nonlinearity. Chen and Li  obtained the existence of infinitely many solutions by using minimax procedure. For more related results, we refer the interested readers to – and references therein.
On the one hand, without , (1.1) reduces to the following Dirichlet problem of Kirchhoff type:
where ρ, , h, E, L are constants, which extends the classical D’Alembert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. Equation (1.2) is related to the stationary analog of problem (1.3). Problem (1.2) received much attention only after Lions  proposed an abstract framework to study the problem. Some important and interesting results can be found; see for example –. We note that results dealing with problem (1.2) with critical nonlinearity are relatively scarce –.
In , by means of a direct variational method, the authors proved the existence and multiplicity of solutions to a class of p-Kirchhoff-type problem with Dirichlet boundary data. In , the authors showed the existence of infinite solutions to the p-Kirchhoff-type quasilinear elliptic equation. But they did not give any further information on the sequence of solutions. Recently, Kajikiya  established a critical point theorem related to the symmetric mountain-pass lemma and applied to a sublinear elliptic equation. However, there are no such results on Kirchhoff-type problems (1.1).
On the other hand, there are many papers concerned with the following quasilinear elliptic equations:
Such equations arise in various branches of mathematical physics and they have been the subject of extensive study in recent years. In , by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (1.4) by the mountain-pass theorem. The same method of a change of variables was used in , but the usual Sobolev space framework was used as the working space and one studied a different class of nonlinearity. In , the existence of both one sign and nodal ground state-type solutions was established by the Nehari method.
Motivated by the reasons above, the aim of this paper is to show the existence of infinitely many soliton solutions of problem (1.1), and there exists a sequence of infinitely many arbitrarily small soliton solutions converging to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya .
Note that behaves like a critical exponent for the above equations; see . For the subcritical case, the existence of solutions for problem (1.4) was studied in – and it was left open for the critical exponent case; see . To the best of our knowledge, the existence of non-trivial radial solutions for (1.4) with was firstly studied by Moameni , where the same Orlicz space as  was used. In , the authors showed the existence of multiple solutions for problems (1.1) with and by minimax methods and the Krasnoselski genus theory. For other interesting results see , .
To the best of our knowledge, the existence and multiplicity of soliton solutions to problem (1.1) has never been studied by variational methods. As we shall see in the present paper, problem (1.1) can be viewed as an elliptic equation coupled with a non-local term. The competing effect of the non-local term with the critical nonlinearity and the lack of compactness of the embedding of into the space prevent us from using the variational methods in a standard way. Some new estimates for such a Kirchhoff equation involving Palais-Smale sequences, which are key points in the application of this kind of theory, need to be established. We mainly follow the idea of , . Let us point out that, although the idea was used before for other problems, the adaptation of the procedure to our problem is not trivial at all; because of the appearance of a non-local term, we must consider our problem for a suitable space and so we need more delicate estimates.
Our main result in this paper is the following.
Suppose thatsatisfies the following conditions:
():, for all;
Then there existssuch that for any, problem (1.1) has a sequence of non-trivial solutionsandinas.
2 Preliminary lemmas
The energy functional corresponding to problem (1.1) is defined as follows:
where for . It should be pointed out that the functional J is not well defined in general, for instance, in . To overcome this difficulty, we employ an argument developed by Colin and Jeanjean . We make the change of variables , where f is defined by
on and by on .
The function f satisfies the following properties:
(f0):f is uniquely definedand invertible.
(f7):The functionis strictly convex.
(f8):There exists a positive constant C such that
(f9):There exist positive constantsandsuch that
So after this change of variables, we can write as
As in , we note that if v is a non-trivial critical point of J, then v is a non-trivial solution of the problem
Therefore, let and since , we conclude that u is a non-trivial solution of the problem
The auxiliary result of this paper is as follows.
Suppose thatsatisfies the following conditions:
(H1):, for all;
Then there existssuch that for any, problem (2.2) has a sequence of non-trivial solutionsandinas.
We recall the second concentration-compactness principle of Lions .
Letbe a weakly convergent sequence to v insuch thatandin the sense of measures. Then, for some at most countable index set I,
where S is the best Sobolev constant, i.e. , , are Dirac measures atand, are constants.
Under assumptions (H1) and (H2), we have
which means that, for all , there exist such that
for some .
Assume conditions (H1) and (H2) hold. Then for any, the functional J satisfies the local (PS) c condition in
in the following sense: if
andfor some sequence in, thencontains a subsequence converging strongly in.
Let be a sequence in such that
Choose , we have and
Thus, we can deduce that . By (2.7) we have
Then by (2.5), we have
Setting , we get
Therefore, the inequalities (2.9) and (2.10) imply that is bounded in . Then is also bounded in . Therefore we can assume that in , a.e. in Ω, since , then a.e. in Ω and then in . Thus, there exist measures μ and ν such that , . Let be a singular point of the measures μ and ν. We define a function such that in , in and in Ω. Let , then is bounded in . Obviously, , i.e.
On the other hand, by the Hölder inequality and (f4) in Lemma 2.1, we have
Similarly, we have
Similar to the proof of (2.12), it follows that
Combining this with Lemma 2.2, we obtain . This result implies that
If the second case holds, for some , then by using the Hölder inequality, we have
By using inequality (2.4), we get
Since , it follows that
By using in the measure sense and Lemma 2.2(i), we have
where . This is impossible. Consequently, for all and hence
Thus, from the weak lower semicontinuity of the norm and we have
since . Thus we prove that strongly converges to v in . □
3 Existence of a sequence of arbitrarily small solutions
In this section, we prove the existence of infinitely many solutions of (1.1) which tend to zero. Let X be a Banach space and denote
For , we define genus as
Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold.
If there exists an odd continuous mapping from A to B, then .
If there is an odd homeomorphism from A to B, then .
If , then .
Then n-dimensional sphere has a genus of by the Borsuk-Ulam theorem.
If A is compact, then and there exists such that and , where .
The following version of the symmetric mountain-pass lemma is due to Kajikiya .
Let E be an infinite-dimensional space andand suppose the following conditions hold.
(C1):is even, bounded from below, andsatisfies the local Palais-Smale condition, i.e. for some, in the case when every sequencein E satisfyingandhas a convergent subsequence.
(C2):For each, there exists ansuch that.
Then either (R1) or (R2) below holds.
(R1):There exists a sequencesuch that, andconverges to zero.
(R1):There exist two sequencesandsuch that, , , , , , , andconverges to a non-zero limit.
From Lemma 3.1 we have a sequence of critical points such that , and .
In order to get infinitely many solutions we need some lemmas. Let , from (2.4) we have
where , , are some positive constants.
Let . Then
Furthermore, there exists such that for , attains its positive maximum, that is, there exists
Therefore, for , we may find such that . Now we define
Then it is easy to see and is . Let and consider the perturbation of :
From the above arguments, we have the following.
Letis defined as in (3.1). Then
and G is even and bounded from below.
If , then , consequently, and .
There exists such that, for , G satisfies a local (PS) c condition for
Assume that (H3) of Theorem 1.1holds. Then for any, there existssuch that.
Firstly, by (H3) of Theorem 1.1, for any fixed , , we have
Secondly, given any , let be a k-dimensional subspace of . Then there exists a constant such that
Therefore for any with and ρ small enough, by (f1) in Lemma 2.1 we have
since . That is,
This completes the proof. □
Now we give the proof of Theorem 1.1.
Proof of Theorem 2.1
By Lemmas 3.2(i) and Lemmas 3.3, we know that . Therefore, assumptions (C1) and (C2) of Lemma 3.1 are satisfied. This means that G has a sequence of solutions converging to zero. Hence, Theorem 2.1 follows by Lemma 3.2(ii). □
Proof of Theorem 1.1
This follows from Theorem 2.1, since if and . □
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The authors would like to express their appreciation of the referees for their precious comments and suggestions as regards the original manuscript. The authors are supported by NSFC (Grant No. 11301038), Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH), Research Foundation during the 12st Five-Year Plan Period of Department of Education of Jilin Province, China (Grant  No. 252), The open project program of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant No. 93K172013K03).
The authors declare that they have no competing interests.
CZ carried out the theoretical studies, and participated in the sequence alignment and drafted the manuscript. YS participated in the design of the study and performed the statistical analysis. SL and FM conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Zhou, C., Miao, F., Liang, S. et al. Multiplicity of solutions for Kirchhoff-type problems involving critical growth. Bound Value Probl 2014, 210 (2014). https://doi.org/10.1186/s13661-014-0210-7
- Kirchhoff-type problems
- critical growth
- concentration-compactness principle
- variational method