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Multiple solutions for the -Laplacian problem involving critical growth with aparameter
Boundary Value Problems volume 2013, Article number: 223 (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.
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.
In this paper, we study the existence and multiplicity of solutions for the quasilinearelliptic problem
where , () is a bounded domain with smooth boundary, is a real parameter, , are continuous functions on with
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:
(f2) there are constants and such that for every , a.e. in Ω,
(f3) there are constants and a continuous function , , with , such that for every , a.e. in Ω,
(f4) there are , and with such that
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.
(g1) there is such that
(g2) , odd with respect to t and
(g3) for all and a.e. in Ω, where .
Moreover, they assumed that
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.
For any , we define the variable exponent Lebesgue space
with the norm
where is the set of all measurable real functions defined onΩ.
Define the space
with the norm
By , we denote the subspace of which is the closure of with respect to the norm . Further, we have
There is a constantsuch that for all,
So, and are equivalent norms in . Hence we will use the norm for all .
Set. For, we have:
Ifwitha.e. in Ω, then thereexists the continuous embedding.
Ifandfor any, the embeddingis compact.
The conjugate space ofis, where. For anyand,
The energy functional corresponding to problem (1.1) is defined on as follows:
Then and ,
We say that is a weak solution of problem (1.1) in the weak sense iffor any ,
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 .
Let, where E is a real Banach spaceand V is finite dimensional. Supposethatis an even functionalsatisfyingand
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.
Let and be two continuous functions such that
Letbe a weakly convergent sequenceinwith weak limit u such that:
weakly in the sense of measures;
weakly in the sense of measures.
Also assume thatis nonempty. Then, for some countableindex set K, we have:
whereand S is the best constant in theGagliardo-Nirenberg-Sobolev inequality for variable exponents,namely
3 Proof of main results
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.
Let be a sequence, i.e., satisfies , and as . If , we have done. So we only need to consider the case that with . We know that
From (f2), we get
Notice that , , then from Lemmas 2.3, 2.4, , so . Let , then , and from the Hölder inequality,
In addition, from Lemma 2.2(2), we can also obtain that
So we have
From (3.1), (3.3) and (f1), we have
Noting that , we have that is bounded.
Up to a subsequence, in .
By Lemma 2.7, we can assume that there exist two measures μ,ν and a function such that
Choose a function such that , on and on . For any , and , let . It is clear that is bounded in . From , we can obtain , as , i.e.,
From (f1), by Lemma 2.7, we have
By the Hölder inequality, it is easy to check that
From (3.5), as , we obtain . From Lemma 2.7, we conclude that
Given , set
where S is given by (2.5). Considering , we have
We claim that . Indeed, if , this follows by (3.7). Otherwise, taking in (3.2), we obtain
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 . □
Next we prove Theorem 1.1 by verifying that the functional satisfies the hypotheses of Lemma 2.6. First, we recallthat each basis for a real Banach space E is a Schauder basis forE, i.e., given , the functional defined by
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.
Proof Now suppose that , with , . From (f3), we know that
Consequently, considering to be chosen posteriorly by Lemma 3.2, we have, for all and j sufficiently large,
Now taking such that and noting that , so , if . We can choose such that . Next, we take such that for ,
for every , , the proof is complete. □
Lemma 3.4 Suppose that f satisfies (f4),then, given, there exist asubspace W ofand a constantsuch thatand.
Proof Let and be such that , and . First, we take with . Considering , we have . Let and such that , and . Next, we take with . After a finite number of steps, we get such that , , and for all . Let , by construction, , and for every ,
consider the case that , then . Now it suffices to verify that
From condition (f4), given , there is such that for every , a.e. x in ,
Consequently, for and ,
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. □
Bocea M, Mihăilescu M: Γ-convergence of power-law functionals with variable exponents. Nonlinear Anal. 2010, 73: 110-121. 10.1016/j.na.2010.03.004
Bocea M, Mihăilescu M, Popovici M: On the asymptotic behavior of variable exponent power-law functionals andapplications. Ric. Mat. 2010, 59: 207-238. 10.1007/s11587-010-0081-x
Bocea M, Mihăilescu M, Pérez-Llanos M, Rossi JD: Models for growth of heterogeneous sandpiles via Mosco convergence. Asymptot. Anal. 2012, 78: 11-36.
Chen Y, Levine S, Rao R: Variable exponent, linear growth functionals in image processing. SIAM J. Appl. Math. 2006, 66: 1383-1406. 10.1137/050624522
Fragnelli G: Positive periodic solutions for a system of anisotropic parabolic equations. J. Math. Anal. Appl. 2010, 367: 204-228. 10.1016/j.jmaa.2009.12.039
Halsey TC: Electrorheological fluids. Science 1992, 258: 761-766. 10.1126/science.258.5083.761
Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 1987, 9: 33-66.
Boureanu MM, Udrea DN:Existence and multiplicity result for elliptic problems with-growth conditions. Nonlinear Anal., Real World Appl. 2013, 14: 1829-1844. 10.1016/j.nonrwa.2012.12.001
Boureanu MM, Preda F: Infinitely many solutions for elliptic problems with variable exponent andnonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 2012, 19(2):235-251. 10.1007/s00030-011-0126-1
Chabrowski J, Fu Y:Existence of solutions for -Laplacian problems on a bounded domain. J. Math. Anal. Appl. 2005, 306: 604-618. 10.1016/j.jmaa.2004.10.028
Dai GW, Liu DH:Infinitely many positive solutions for a -Kirchhoff-type equation involving the-Laplacian. J. Math. Anal. Appl. 2009, 359: 704-710. 10.1016/j.jmaa.2009.06.012
Fan XL, Zhang QH:Existence of solutions for -Laplacian Dirichlet problem. Nonlinear Anal. 2003, 52: 1843-1852. 10.1016/S0362-546X(02)00150-5
Mihăilescu M:On a class of nonlinear problems involving a -Laplace type operator. Czechoslov. Math. J. 2008, 58(133):155-172.
Bonder JF, Silva A: The concentration compactness principle for variable exponent spaces andapplications. Electron. J. Differ. Equ. 2010., 2010: Article ID 141
Fu YQ:The principle of concentration compactness in spaces and its application. Nonlinear Anal. 2009, 71: 1876-1892. 10.1016/j.na.2009.01.023
Silva, A: Multiple solutions for the -Laplace operator with critical growth. Preprint
Alves CO, Barrwiro JLP:Existence and multiplicity of solutions for a -Laplacian equation with critical growth. J. Math. Anal. Appl. 2013, 403: 143-154. 10.1016/j.jmaa.2013.02.025
Fu YQ, Zhang X:Multiple solutions for a class of -Laplacian equations in involving the critical exponent. Proc. R. Soc., Math. Phys. Eng. Sci. 2010, 466(2118):1667-1686. 10.1098/rspa.2009.0463
Bonder, JF, Saintier, N, Silva, A: On the Sobolev trace theorem for variableexponent spaces in the critical range. Preprint
Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7
Silva EAB, Xavier MS: Multiplicity of solutions for quasilinear elliptic problems involving criticalSobolev exponents. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2003, 20(2):341-358. 10.1016/S0294-1449(02)00013-6
Fan X, Zhao D:On the space and . J. Math. Anal. Appl. 2001, 263: 424-446. 10.1006/jmaa.2000.7617
Kovacik O, Rakosnik J:On spaces and . Czechoslov. Math. J. 1991, 41: 592-618.
Lindenstrauss J, Tzafriri L: Classical Banach Spaces, I. Springer, Berlin; 1977.
Marti JT: Introduction to the Theory of Bases. Springer, New York; 1969.
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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.