Positive symmetric solutions for a class of critical quasilinear elliptic problems in
© Deng and Huang; licensee Springer. 2014
Received: 12 November 2013
Accepted: 5 June 2014
Published: 24 September 2014
This paper deals with the critical quasilinear elliptic problem in , where is the p-Laplacian, , with , , , and Q and h are measurable functions satisfying some symmetry conditions with respect to a closed subgroup G of . By variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of positive G-symmetric solutions under certain appropriate hypotheses on Q, h, and q.
MSC: 35J25, 35J60, 35J65.
where () is a smooth domain (bounded or unbounded) containing the origin, , is the critical Sobolev exponent, and is a measurable function with subcritical growth. The main reason of interest in singular potentials relies in their criticality: they have the same homogeneity as the Laplacian and the critical Sobolev exponent and do not belong to the Kato class, hence they cannot be regarded as the lower order perturbation terms. We also mention that (1.1) is related to applications in many physical contexts: fluid mechanics, glaciology, molecular physics, quantum cosmology and linearization of combustion models (see  for example). So for this reason, many existence, nonexistence, and multiplicity results for equations like (1.1) have been obtained with various hypotheses on the measurable function ; we refer the readers to – and the references therein. Moreover, for other results on this aspect, see  for boundary singularities,  for high-order nonlinearity,  for non-autonomous Schrödinger-Poisson systems in ,  for singular elliptic systems in , and  for large singular sensitivity etc.
where , , are constants, , , and k fulfills certain symmetry conditions with respect to a subgroup G of . By the concentration-compactness principle in Refs. ,  and variational methods, the authors obtained the existence and multiplicity of G-symmetric solutions under some assumptions on k. Very recently, Deng and Huang – extended the results in Ref.  to nonlinear singular elliptic problems in a bounded G-symmetric domain. We also mention that when and the right-hand side term is replaced by ( or ) in (1.2), the existence and multiplicity of G-symmetric solutions of (1.2) were obtained in Refs. –. Finally, when , we remark that Su and Wang  established the existence of nontrivial radial solutions for a class of quasilinear singular equations such as (1.2) by proving several new embedding theorems.
where is the p-Laplacian, , , with , , is the critical Hardy-Sobolev exponent and is the critical Sobolev exponent; Q and h are G-symmetric functions (see Section 2 for details) satisfying some appropriate conditions which will be specified later. Problem (1.3) is in fact a continuation of (1.2). However, due to the nonlinear perturbation and the singularities caused by the terms and , compared with the semilinear equation (1.2), the critical quasilinear equation (1.3) becomes more complicated to deal with and we have to overcome more difficulties in the study of G-symmetric solutions. As far as we know, there are few results on the existence of G-symmetric solutions for (1.3) as , , and . Hence, it makes sense for us to investigate problem (1.3) thoroughly. Let be a constant. Note that here we will try to treat both the cases of , , and , .
This paper is organized as follows. In Section 2, we will establish the appropriate Sobolev space which is applicable to the study of problem (1.3), and we will state the main results of this paper. In Section 3, we detail the proofs of some existence and multiplicity results for the cases and in (1.3). In Section 4, we give the proofs of existence results for the cases and in (1.3). Our methods in this paper are mainly based upon the symmetric criticality principle of Palais (see ) and variational arguments.
2 Preliminaries and main results
Let be the group of orthogonal linear transformations of with natural action and let be a closed subgroup. For we denote the cardinality of by and set . Note that here may be +∞. For any function , We call a G-symmetric function if for all and , holds. In particular, if f is radially symmetric, then the corresponding group G is and . Other further examples of G-symmetric functions can be found in Ref. .
By the Hardy inequality (2.2), we easily see that the above norm is equivalent to the usual norm .
We suppose that and fulfill the following conditions.
(q.1), and is G-symmetric.
(q.2), where .
(h.1) is G-symmetric.
(h.2) is nonnegative and locally bounded in , in the bounded neighborhood of the origin, as , , , where .
The main results of this paper are the following.
for some, where, then problem () has at least one positive solution in.
- (1)Problem () has a positive solution if
- (2)Problem () admits at least one positive solution if exists and is positive,
- (3)If on and
then problem () has at least one positive solution.
Suppose thatand. Then problem () has infinitely many G-symmetric solutions.
then problem () possesses at least one positive solution in.
Throughout this paper, we denote by the subspace of consisting of all G-symmetric functions. The dual space of (, resp.) is denoted by (, resp.), where . In a similar manner, we define for an open and G-symmetric subset of , that is, if , then for all . In the case where Ω is bounded, we set . The ball of center x and radius r is denoted by . We employ C, () to denote the positive constants, and denote by ‘→’ convergence in norm in a given Banach space X and by ‘⇀’ weak convergence. A functional is said to satisfy the condition if each sequence in X satisfying , in has a subsequence which strongly converges to some element in X. Hereafter, denotes the weighted space with the norm . Also, for nonnegative measurable function , we denote by the space of measurable functions u satisfying .
3 Existence and multiplicity results for problem ()
Letbe a G-symmetric function; inimpliesin.
where, , is the Dirac mass of 1 concentrated at.
To find critical points of ℱ we need the following local condition which is crucial for the proof of Theorem 2.1.
which is impossible. Hence for all , and consequently we have in . Finally, observe that and, thus by we obtain in . □
As an easy consequence of Lemma 3.3 we obtain the following result.
Ifand, then the functional ℱ satisfiescondition for every.
Proof of Theorem 2.1
If , then by Lemma 3.3, the condition holds and the conclusion follows from the mountain pass theorem in Ref.  (see also ). If , then , with , is a path in Γ such that . Consequently, either and we are done, or γ can be deformed to a path with and we get a contradiction. This part of the proof shows that a nontrivial solution of () exists. We now show that the solution can be chosen to be positive on . Since and , we obtain , which implies . Hence, either is a critical point of ℱ or , with , can be deformed, as above of the proof, to a path with , which is impossible. Therefore, we may assume that is nonnegative on and the fact that on follows by the strong maximum principle. □
Proof of Corollary 2.1
for some constant independent of ϵ. Combining (3.12) and (3.13), we get (3.11) for ϵ sufficiently small.
Thus (3.11) holds for ϵ sufficiently small.
for some constant independent of . These two estimates combined together give (3.14) for large.
and (3.14) holds for large. Similarly to above, we find that part (3) holds. □
To prove Theorem 2.2 we need the following version of the symmetric mountain pass theorem (cf., Theorem 9.12]).
there exist constants and such that for all ;
there exists an increasing sequence of subspaces of E, with , such that for every m one can find a constant such that for all with .
Then ℱ possesses a sequence of critical valuestending to ∞ as.
Proof of Theorem 2.2
for t large enough. By Corollary 3.1 and Lemma 3.4 we conclude that there exists a sequence of critical values as and the result follows. □
4 Existence results for problem ()
The aim of this section is to discuss problem () and prove Theorem 2.3; here is a constant. First, we give the following compact embedding result which is indispensable for the proof of Theorem 2.3.
Suppose that (h.2) holds. Thenis compactly embedded in. Furthermore, if h satisfies (h.1) and (h.2) andis closed, then the inclusion ofinis compact.
as . Suppose that and is bounded in . We may assume in . By the compactness of the inclusion of in and the local boundedness of , we easily see that . Therefore, by taking and , we conclude from (4.1) and (4.2) that . This implies the compactness of the inclusion of in .
On the other hand, since is closed and is a compact Lie group, G is compact. Consequently, by using the first part of the proof and the methods in Schneider , Corollaries 3.4 and 3.2], we deduce that is compactly embedded in and the results follow. □
where . By (2.1), (h.1), (h.2), and Lemma 4.1, we easily see that is well defined and of . Thus there exists a one-to-one correspondence between the weak solutions of () and the critical points of . Moreover, an analogously symmetric criticality principle of Lemma 3.1 clearly holds; hence the weak solutions of problem () are exactly the critical points of .
Therefore we conclude that satisfies (4.9) for sufficiently small and the result follows. □
which contradicts (4.14). Therefore, we obtain as , and hence, in . The proof of this lemma is completed. □
Proof of Theorem 2.3
Combining the above inequality and Lemma 4.3, we obtain a critical point of satisfying (). Taking as the test function, we get . This implies in . By the strong maximum principle, we obtain in . This, combined with the symmetric criticality principle, implies that is a positive G-symmetric solution of (). □
The authors express their thanks to the referees for their valuable comments and suggestions. This work is supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ130503), and it is partially supported by the Natural Science Foundation of China (Grant No. 11171247).
- Dautray R, Lions JL: Mathematical analysis and numerical methods for science and technology. In Physical Origins and Classical Methods. Springer, Berlin; 1990.Google Scholar
- Garcia Azorero J, Peral I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 1998, 144: 441-476. 10.1006/jdeq.1997.3375MathSciNetView ArticleGoogle Scholar
- Ghoussoub N, Yuan C: Multiple solutions for quasilinear PDEs involving critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 2000, 352: 5703-5743. 10.1090/S0002-9947-00-02560-5MathSciNetView ArticleGoogle Scholar
- Kang DS, Deng YB: Existence of solution for a singular critical elliptic equation. J. Math. Anal. Appl. 2003, 284: 724-732. 10.1016/S0022-247X(03)00394-9MathSciNetView ArticleGoogle Scholar
- Kang DS: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 2008, 68: 1973-1985. 10.1016/j.na.2007.01.024MathSciNetView ArticleGoogle Scholar
- Chen ZJ, Zou WM: On an elliptic problem with critical exponent and Hardy potential. J. Differ. Equ. 2012, 252: 969-987. 10.1016/j.jde.2011.09.042MathSciNetView ArticleGoogle Scholar
- Deng YB, Li Y, Yang F: On the positive radial solutions of a class of singular semilinear elliptic equations. J. Differ. Equ. 2012, 253: 481-501. 10.1016/j.jde.2012.02.017MathSciNetView ArticleGoogle Scholar
- Deng YB, Jin LY, Peng SJ: Solutions of Schrödinger equations with inverse square potential and critical nonlinearity. J. Differ. Equ. 2012, 253: 1376-1398. 10.1016/j.jde.2012.05.009MathSciNetView ArticleGoogle Scholar
- Shang YY: Existence and multiplicity of positive solutions for some Hardy-Sobolev critical elliptic equation with boundary singularities. Nonlinear Anal. 2012, 75: 2724-2734. 10.1016/j.na.2011.11.013MathSciNetView ArticleGoogle Scholar
- Waliullah S: Higher order singular problems of Caffarelli-Kohn-Nirenberg-Lin type. J. Math. Anal. Appl. 2012, 385: 721-736. 10.1016/j.jmaa.2011.07.005MathSciNetView ArticleGoogle Scholar
- Sun JT, Chen HB, Nieto JJ: On ground state solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 2012, 252: 3365-3380. 10.1016/j.jde.2011.12.007MathSciNetView ArticleGoogle Scholar
- De Souza M:On a singular class of elliptic systems involving critical growth in . Nonlinear Anal., Real World Appl. 2011, 12: 1072-1088. 10.1016/j.nonrwa.2010.09.001MathSciNetView ArticleGoogle Scholar
- Stinner C, Winkler M: Global weak solutions in a chemotaxis system with large singular sensitivity. Nonlinear Anal., Real World Appl. 2011, 12: 3727-3740.MathSciNetGoogle Scholar
- Deng YB, Jin LY: On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent. J. Math. Anal. Appl. 2007, 329: 603-616. 10.1016/j.jmaa.2006.06.070MathSciNetView ArticleGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations. The limit case. Rev. Mat. Iberoam. 1985, 1: 145-201. (part 1) 10.4171/RMI/6View ArticleGoogle Scholar
- Lions PL: The concentration-compactness principle in the calculus of variations. The limit case. Rev. Mat. Iberoam. 1985, 1: 45-121. (part 2) 10.4171/RMI/12View ArticleGoogle Scholar
- Deng ZY, Huang YS: On G -symmetric solutions of a quasilinear elliptic equation involving critical Hardy-Sobolev exponent. J. Math. Anal. Appl. 2011, 384: 578-590. 10.1016/j.jmaa.2011.06.012MathSciNetView ArticleGoogle Scholar
- Deng ZY, Huang YS: Existence and multiplicity of symmetric solutions for a class of singular elliptic problems. Nonlinear Anal., Real World Appl. 2012, 13: 2293-2303. 10.1016/j.nonrwa.2012.01.024MathSciNetView ArticleGoogle Scholar
- Deng ZY, Huang YS: Existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical Hardy-Sobolev exponents. J. Math. Anal. Appl. 2012, 393: 273-284. 10.1016/j.jmaa.2012.04.011MathSciNetView ArticleGoogle Scholar
- Bianchi G, Chabrowski J, Szulkin A: On symmetric solutions of an elliptic equations with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal. 1995, 25: 41-59. 10.1016/0362-546X(94)E0070-WMathSciNetView Article