- Open Access
Multiple solutions for the -Laplacian problem involving critical growth with aparameter
© Yang et al.; licensee Springer. 2013
- Received: 11 April 2013
- Accepted: 5 September 2013
- Published: 7 November 2013
By energy estimates and establishing a local condition, existence of solutions for the-Laplacian problem involving critical growth in abounded domain is obtained via the variational method under the presence ofsymmetry.
MSC: 35J20, 35J62.
- -Laplacian problem
- critical Sobolev exponents concentration-compactness principle
In recent years, the study of problems in differential equations involving variableexponents has been a topic of interest. This is due to their applications in imagerestoration, mathematical biology, dielectric breakdown, electrical resistivity,polycrystal plasticity, the growth of heterogeneous sand piles and fluid dynamics,etc. We refer readers to [1–7] for more information. Furthermore, new applications are continuing to appear,see, for example,  and the references therein.
With the variational techniques, the -Laplacian problems with subcritical nonlinearities havebeen investigated, see [9–13]etc. However, the existence of solutions for -Laplacian problems with critical growth is relatively new.In 2010, Bonder and Silva  extended the concentration-compactness principle of Lions to the variableexponent spaces, and a similar result can be found in . After that, there have been many publications for this case, see [16–19]etc.
Related to f, we assume that is a Carathéodory function satisfying for every , and the subcritical growth condition:
(f1) for all , where is a continuous function in satisfying , .
For , we suppose that f satisfies the following:
Now we state our result.
Theorem 1.1 Assume that (1.2), (1.3) and(f1)-(f4) are satisfied with, is odd in s. Then,given, there existssuch that problem (1.1) possesses atleast k pairs of nontrivial solutions forall.
(g3) for all and a.e. in Ω, where .
and the result is the following theorem.
Theorem 1.2 Assume that (1.2), (1.3), (1.4) and(g1)-(g3) are satisfied with. Then there exists asequencewithsuch that for, problem (1.1) has atleast k pairs of nontrivial solutions.
Note that (f2) is a weaker version of (g3). This conditioncombined with (f1) and the concentration-compactness principle in  will allow us to verify that the associated functional satisfies the condition  below a fixed level for sufficiently small. Conditions (f3) and(f4) provide the geometry required by the symmetric mountain pass theorem . Compared with (g2), there is no condition imposed on fnear zero in Theorem 1.1. Furthermore, we should mention that our Theorem 1.1 improvesthe main result found in . In that paper, the authors considered only the case where is constant, while in our present paper, we have showedthat the main result found in  is still true for a large class of functions.
The paper is organized as follows. In Section 2, we introduce some necessarypreliminary knowledge. Section 3 contains the proof of our main result.
We recall some definitions and basic properties of the generalized Lebesgue-Sobolevspaces and , where is a bounded domain with smooth boundary. And Cwill denote generic positive constants which may vary from line to line.
where is the set of all measurable real functions defined onΩ.
By , we denote the subspace of which is the closure of with respect to the norm . Further, we have
So, and are equivalent norms in . Hence we will use the norm for all .
Ifwitha.e. in Ω, then thereexists the continuous embedding.
Ifandfor any, the embeddingis compact.
So, the weak solution of problem (1.1) coincides with the critical point of. Next, we need only to consider the existence of criticalpoints of .
We say that satisfies the condition if any sequence , such that and as , possesses a convergent subsequence. In this article, weshall be using the following version of the symmetric mountain pass theorem .
there is a constantsuch that;
there is a subspace W of E withand there issuch that;
consideringgiven by (ii), I satisfiesfor.
Then I possesses at leastpairs of nontrivial critical points.
Next we would use the concentration-compactness principle for variable exponent spaces.This will be the keystone that enables us to verify that satisfies the condition.
Letbe a weakly convergent sequenceinwith weak limit u such that:
weakly in the sense of measures;
weakly in the sense of measures.
Lemma 3.1 Assume that f satisfies (f1)and (f2) with. Then, given, there existssuch thatsatisfies thecondition for all, provided.
Proof (1) The boundedness of the sequence.
Up to a subsequence, in .
Therefore, by (3.8), the claim is proved. As a consequence of this fact, we concludethat for all . Therefore, in . Then, with the similar step in , we can get that in . □
Lemma 3.2 Givenfor alland, there issuch that for all, .
Proof We prove the lemma by contradiction. Suppose that there exist and for every such that . Taking , we have for every and . Hence is a bounded sequence, and we may suppose, without loss ofgenerality, that in . Furthermore, for every since for all . This shows that . On the other hand, by the compactness of the embedding, we conclude that . This proves the lemma. □
Lemma 3.3 Suppose that f satisfies (f3),then there existandsuch thatfor all.
for every , , the proof is complete. □
Lemma 3.4 Suppose that f satisfies (f4),then, given, there exist asubspace W ofand a constantsuch thatand.
where and . Observing that W is finite dimensional, we have, , and the inequality is obtained by taking. The proof is complete. □
Proof of Theorem 1.1 First, we recall that , where and are defined in (3.9). Invoking Lemma 3.3, we find, and satisfies (i) with . Now, by Lemma 3.4, there is a subspace W of with such that satisfies (ii). By Lemma 3.1, satisfies (iii). Since and is even, we may apply Lemma 2.6 to conclude that possesses at least k pairs of nontrivial criticalpoints. The proof is complete. □
The authors would like to express their gratitude to the anonymous referees forvaluable comments and suggestions which improved our original manuscript greatly. Thefirst author is supported by NSFC-Tian Yuan Special Foundation (No. 11226116),Natural Science Foundation of Jiangsu Province of China for Young Scholar (No.BK2012109), the China Scholarship Council (No. 201208320435), the FundamentalResearch Funds for the Central Universities (No. JUSRP11118, JUSRP211A22). The secondauthor is supported by NSFC (No. 10871096). The third author is supported by GraduateEducation Innovation of Jiangsu Province (No. CXZZ13-0389).
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