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Deng, Zhiying; Huang, Yisheng

doi:https://doi.org/10.1155/2014/430976

Abstract and Applied Analysis

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Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 430976 | https://doi.org/10.1155/2014/430976

Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents in

Zhiying Deng¹and Yisheng Huang²

Academic Editor: Márcia Federson

Received02 Apr 2014

Accepted28 Aug 2014

Published16 Nov 2014

Abstract

This paper deals with a class of quasilinear elliptic systems involving singular potentials and critical Sobolev exponents in . By using the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the potentials and parameters.

1. Introduction

In this work, we investigate the existence and multiplicity of nontrivial solutions for the following quasilinear elliptic system: where is a quasilinear elliptic operator, , , with , , and satisfy , denotes the critical Sobolev exponent, and and with are -symmetric functions (see Section 2 for details) with respect to a closed subgroup of .

In recent years, considerable attention has been paid to the scalar singular elliptic problem: where is a smooth domain (bounded or unbounded) containing the origin. The study of this type of equation is motivated by its definite physics background and various applications, including celestial mechanics, fluid mechanics, and flow through porous media (see [1]). The mathematical interest lies in the fact that these problems like (2) are doubly critical due to the presence of the Sobolev embedding and the singularities. For this reason, many existence, nonexistence, and multiplicity results of nontrivial solutions for the single equations like (2) have been established with different assumptions on the potentials , and the parameters , , and ; we refer to [2–8] and the references therein.

In a recent paper, Deng and Jin [9] considered the following single semilinear elliptic problem: where , , , , and satisfies some symmetry conditions with respect to . By using analytic techniques and variational arguments, the authors proved the existence and multiplicity of -symmetric solutions to (3) under certain hypotheses on . Subsequently, Waliullah [10] improved the results in [9] by using the minimizing sequence and the concentration-compactness principle. Recently, Deng and Huang [11] extended the results in [9, 10] to the scalar weighted elliptic problems in a bounded -symmetric domain. Besides these, when and the right-hand side term is replaced by a term of the pure power, such as with or , there are many interesting results on the existence and multiplicity of -symmetric solutions of (3), which can be found in [12–14] and the references therein.

On the other hand, there have been many papers concerned with the existence and multiplicity of nontrivial solutions for elliptic systems. In [15], Wu considered the following semilinear elliptic system: where is a smooth bounded domain, , , , , and the weight functions , , fulfill certain suitable conditions. Via the analytic techniques of Nehari manifold and variational methods, the author proved that the system (4) admits at least two nontrivial nonnegative solutions if the pair of parameters belongs to a certain subset of . Very recently, Nyamoradi [16], Lü and Xiao [17], and Li and Gao [18] generalized the corresponding results of [15] to the nonlinear singular elliptic systems involving critical Hardy-Sobolev exponents. Other results about existence and multiplicity of nontrivial solutions, also for related elliptic systems, can be seen in [19–23] and the references therein.

However, as far as we know, the existence and multiplicity of -symmetric solutions for singular elliptic systems were seldom studied; we can only find some -symmetric results for singular elliptic systems in [24] and, when , some radial and nonradial results for nonsingular elliptic systems in [25]. Inspired by [9, 12, 25], in this paper we are concerned with the existence and multiplicity of positive -symmetric solutions for system (1). The main difficulties lie in the fact that there are not only the nonlinear perturbations , and the Hardy singular terms , in (1), but also four nonlinear terms with the critical Sobolev exponents in . Compared with (3) and (4), the singular quasilinear elliptic system (1) becomes more complicated to deal with. Moreover, the approach involving the Nehari manifold requires that the corresponding nonlinearity is second order derivative about and . Hence, in order to obtain the multiple -symmetric solutions of system (1), the Nehari manifold techniques in the literature mentioned above are invalid and we need to look for other methods. To our knowledge, even in the particular case and , there are no results on the existence and multiplicity of -symmetric solutions for system (1). It is therefore meaningful for us to investigate system (1) deeply. Let be a constant. Note that, here, we will try to treat both the cases of , and , .

This paper is schemed as follows. In Section 2, we establish the appropriate Sobolev space which is applicable to the study of the elliptic system (1) and state the main results of this paper. In Section 3, we detail the proofs of several existence and multiplicity results for the case and in (1). In Section 4, we will present the proofs of multiplicity results for the case and in (1). Our methods in this paper are mainly based upon the symmetric criticality principle of Palais (see [26]) and variational arguments.

2. Preliminaries and Main Results

Let be the group of orthogonal linear transformations in and let be a closed subgroup. For we denote the cardinality of by and set . Note that, here, may be . We say that is -symmetric (or -invariant) if for every and and in the context of Sobolev spaces this equality is understood a.e. on . In particular, if is radially symmetric, then the corresponding group is and . We call a -symmetric subset of ; if , then for all .

Let denote the closure of functions with respect to the norm . We recall that the well-known Hardy inequality (see [2, 3]) holds: where . For , we employ the following norm in : By inequality (5), we see that the above norm is equivalent to the usual norm . The elliptic operator is positive in if . Moreover, we define the product space endowed with the norm The natural functional space to study system (1) is the Banach space , which is the subspace of consisting of all -symmetric functions. Now in this paper, we are concerned with the following elliptic problems: To mention our main results, we need to introduce two notations and , which are, respectively, defined by where , , and the constant , depending only on , , and . From Kang [4], we see that satisfies the equations The function in (10) is radially symmetric. Moreover, the following asymptotic properties at the origin and infinity for and hold [4]: where , are positive constants and and are the zeroes of the function satisfying

We suppose that the functions , , and verify the following hypotheses.(q.1), and is -symmetric.(q.2), where .(h.1) and are -symmetric.(h.2), , and with , where .

The main results of this paper are summarized in the following.

Theorem 1. Suppose that (q.1) and (q.2) hold. If for some , where , then problem has at least one positive solution in .

Corollary 2. Suppose that (q.1) and (q.2) hold. Then we have the following statements. (1)Problem has a positive solution ifand either (i) for some and small or (ii) for some constant , and small and(2)Problem admits at least one positive solution if exists and is positive,and either (i) for some and large or (ii) for some constants , and large and(3)If on and then problem has at least one positive solution.

Theorem 3. Suppose that and . Then problem has infinitely many -symmetric solutions.

Corollary 4. If is a radially symmetric function such that , then problem has infinitely many solutions which are radially symmetric.

Theorem 5. Let be a constant. Suppose that and (h.1), (h.2) hold. Then there exists such that, for any , problem possesses at least two positive solutions in .

Remark 6. The main results of this paper generalize, extend, and complement some results of the aforementioned papers [9–12, 24, 25].

In the sequel, we denote by the subspace of consisting of all -symmetric functions. The dual space of (, resp.) is denoted by (, resp.), where . The ball of center and radius is denoted by . We employ to denote (possibly different) positive constants and denote by “” convergence in norm in a given Banach space and by “⇀” weak convergence. Hereafter, denotes a datum which tends to as . denotes the weighted space with the norm . A functional is said to satisfy the condition if each sequence in satisfying , in has a subsequence which strongly converges to some element in .

3. Existence and Multiplicity Results for Problem

The corresponding energy functional of problem is defined in by Note that and (5) imply that . It is well known that there exists a one-to-one correspondence between the weak solutions of problem and the critical points of . More precisely, any weak solution of is exactly the critical point of by the following symmetric principle (see Lemma 7); namely, satisfies if and only if for all , there holds

Lemma 7. in implies in .

Proof. See the proof of [12, Lemma 1] (see also [25, Proposition 2.8]).

Now, for any , , , , and , we define where is a minimal point of and therefore a root of the equation

Lemma 8. Suppose that , , and . Then , and has the minimizer , , where is the extremal function of defined as in (10).

Proof. The proof is similar to the proof in Nyamoradi [16, Theorem 2].

Lemma 9. Let be a weakly convergent sequence to in such that , , , , , , and in the sense of measures. Then there exists some at most countable set , , , , , , , , and such that
(a), ,(b), ,(c), , ,(d), , ,(e), , , where , , is the Dirac-mass of 1 concentrated at .

Proof. The proof is similar to that of the concentration compactness principle in [27, 28] (see also [20, Lemma 2.2]) and is omitted here.

In order to find critical points of , we need the following local condition.

Lemma 10. Suppose that (q.1) and (q.2) hold. Then the condition in holds for if

Proof. The proof is similar to that in [12, Proposition 2]. We sketch the argument here for completeness. Suppose satisfies and with . It is easy to show that is bounded in and then up to a subsequence. Moreover, we know from Lemma 9 that there exist measures , , , , , , and such that relations (a)–(e) of this lemma hold. Let be a singular point of measures , , and . As in [20], we can choose two functions such that , , for , for and , . By Lemma 7, , and, hence, using the Sobolev inequality and the Hölder inequality, we have Taking limits as in (30), we obtain from Lemma 9 and the fact that that The above inequality implies that the concentration of the measures , , and cannot occur at points where ; that is, if then . Combining (31) and (d) of Lemma 9 we infer that either (i) or (ii) . For the point , similarly as in the case , we get This, combined with (e) of Lemma 9, implies that either (iii) or (iv) . To study the concentration at infinity of the sequence we need to consider the following quantities: (1), ,(2), ,(3),(4), . Obviously, , , , , and exist and are finite. For , let and be two regular functions such that , , for , for and , . Since the sequence is bounded in , we get from (23) that We now observe that in . Therefore, using the Sobolev inequality and the Hölder inequality we can easily check that Consequently, taking into account the definitions (1)–(4) of , , , , , , and , we deduce from (33) and (34) that On the other hand, by (5) and the definition (25) of we easily see that , and This, combined with (35), implies that either () or (vi) . We now rule out the cases (ii), (iv), and (vi). For every continuous nonnegative function such that on , we obtain from (23) and (24) that If (ii) occurs, then the set must be finite because the measures , , and are bounded. Since functions are -symmetric, the measures , , and must be -invariant. This means that if is a singular point of , , and , so is for each , and the mass of , , and concentrated at is the same for each . If we assume the existence of with such that (ii) holds, then we choose with compact support so that for each and we obtain which contradicts (29). Similarly, if (iv) holds for , we choose with compact support, so that and we have which is impossible. Finally, if (vi) occurs at , we take to get a contradiction with (29). Consequently, for all and this implies that Finally, observe that and, hence, by , we obtain as in . The assertion follows.

As an immediate consequence of Lemma 10 we obtain the following result.

Corollary 11. If and , then the functional satisfies the condition for every .

Proof of Theorem 1. Firstly, we choose such that the condition (17) holds, where is the extremal function satisfying (10), (11), and (12). By (q.1), (23), and (25), we have Hence there exist constants and such that for all . Furthermore, if we set , and with , then we can check that has a unique maximum at some . A simple computation gives us the value Consequently, we obtain from (26) and (27) that Since as , we can choose such that and and set where From (11), (17), (29), (45), (46), and Lemma 8, we obtain thatIf , then we conclude from Lemma 10 that the condition holds and the conclusion follows by the mountain pass theorem in [29] (see also [30]). If , then , with , is a path in such that . Thus, either and we are done, or can be deformed to a path with , which is impossible. Hence we have a nontrivial solution to problem . In the following, we have just to show that the solution can be chosen to be positive on . Consider the Nehari manifold Writing an arbitrary element as , with , we deduce from (24), (25), and the fact that that This implies that , with a constant independent of . Thus we conclude that the set is bounded away from and . Set Then , and, hence, is a -manifold. Notice that and set . We now claim . Indeed, if , then we can find such that . Consequently, we deduce that By a straightforward calculation, we get Setting for , with such that , we obtain that on , a contradiction with the definition (46) of . Hence we have . Finally, by the strong maximum principle, we obtain and on . This, combined with (24) and Lemma 7, implies that is a positive -symmetric solution of .

Proof of Corollary 2. First of all, we observe that, due to the identity (12), inequality (17) is equivalent to for some , or equivalently for some , where and . Part (1), case (i): according to (54), we need to show that for some . We choose so that for . This, combined with (14) and (16) and the fact that , implies that as . On the other hand, for all , we have for some constant independent of . Combining (56) and (57), we get (55) for sufficiently small.
Part (1), case (ii): we choose so that for . Since , we deduce from (14) and the fact that that So by (14), (16), (19), and the Lebesgue dominated convergence theorem we obtain that Hence, (55) holds for sufficiently small.
Part (2), case (i): from (54) it is sufficient to show that for some . We choose such that for all . Then as . Moreover, in view of and , we obtain for some constant independent of . These two estimates combined together give (60) for large.
Part (2), case (ii): we choose such that for all . Since and , we get Consequently, by (13), (16), (21), and the Lebesgue dominated convergence theorem, we have and (60) holds for large. Similarly as above, we know part (3) holds.

To prove Theorem 3 we need the following version of the symmetric mountain pass theorem (cf. [31, Theorem 9.12]).

Lemma 12. Let be an infinite dimensional Banach space and let be an even functional satisfying condition for each and . Furthermore, one supposes that (i)there exist constants and such that for all ;(ii)there exists an increasing sequence of subspaces of , with , such that for every one can find a constant such that for all with . Then possesses a sequence of critical values tending to as .

Proof of Theorem 3. We follow the arguments of [12]. Applying Lemma 12 with and , we see from (q.1), (23), and (25) that Since , there exist constants and such that for all with . To find a suitable sequence of finite dimensional subspaces of , we set . Obviously, the set is -symmetric and we can define , which is the subspace of -symmetric functions of (see Section 2). By extending functions in by outside we can assume that . Let be an increasing sequence of subspaces of with for each . Then we deduce that there exists a constant such that for all , with . Consequently, if then we write , with and . Hence we obtain for large enough. Therefore we conclude from Lemma 12 and Corollary 11 that there exists a sequence of critical values and the results follow.

Proof of Corollary 4. Since is radially symmetric, that is, , we easily see that the corresponding group and . According to Corollary 11, satisfies the condition for every . Hence, by applying the proof of Theorem 3 the conclusion follows.

4. Multiplicity Results for Problem

Throughout this section we assume that and is a constant. Since we are interested in positive -symmetric solutions of problem , we define a functional given by where , and . By (h.1), (h.2), and the Hölder inequality, we easily see that and there exists a one-to-one correspondence between the weak solutions of and the critical points of . Furthermore, an analogously symmetric principle of Lemma 7 clearly holds; hence, the weak solutions of problem are exactly the critical points of the functional .

Lemma 13. Suppose that (h.1) and (h.2) hold. Then there exists a positive constant depending on , , , , , and , such that any bounded sequence , satisfying contains a convergent subsequence.

Proof. Since is bounded in , we can obtain a subsequence, still denoted by , satisfying Moreover, using (h.2) and the Hölder inequality and the Lebesgue dominated theorem, we may also assume as . By (71) and the standard argument, we easily show that is a critical point of . Consequently, we deduce from (68), (9), (h.2), the Hölder inequality, and the fact that that where is a positive constant. Now we set and . Then by the Brezis-Lieb lemma [32] and arguing as in [33, Lemma 2.1] we get Since and , we obtain from (68), (71), and (73) that Hence, for a subsequence , we have as . From the definition (25) of it follows that , which implies either or . If , we obtain from (72) and (74) that which contradicts (69). Consequently, we have as , and, thus, in . The lemma is proved.

Lemma 14. Suppose that (h.1) and (h.2) hold. Then there exists such that for any the following geometric conditions for hold: (i); there exist and such that for all ;(ii)there exists such that and .

Proof. According to (h.1) and (h.2), for all , we deduce from (9), (25), (68), the Young inequality, and the Hölder inequality that where is a constant depending on . The last inequality and the fact imply that, for small , there exist constants , , and such that for all and . On the other hand, since , we conclude from (68) that there exists such that as , which completes this proof.

Lemma 15. Suppose that (h.1) and (h.2) hold. Then there exists such that for any and small , where satisfies (26)–(28) and is given in Lemma 13.

Proof. First, we define the functions with . Note that , for , and . Hence can be achieved at some finite at which becomes zero. By direct calculation, we obtain from (11), (12), (26), (27), (80), and Lemma 8 that Let be such that , . Then from (h.1), (h.2), (11), and (79), we have and there exists independent of such that Moreover, we obtain from (79)–(81) that Now, taking such that that is, we have Choosing , we deduce from (83) and (87) that Therefore the result of this lemma follows.

Proof of Theorem 5. Taking and , for , given in the proofs of Lemmas 14 and 15, we define where . Since the metric space is complete, we deduce from the Ekeland variational principle [34] that there exists a sequence such that and as . Let be the -symmetric functions such that . By (h.1) and (h.2), we have . This, combined with the fact that , implies that there exists sufficiently small such that Therefore we obtain for any . By Lemma 13, possesses a critical point with . Taking as a pair of test functions, where and , we deduce from (68) that , which implies and in . Consequently, by the strong maximum principle and the symmetric criticality principle, we conclude that is a positive -symmetric solution of problem .
On the other hand, we define where . It follows from Lemmas 14 and 15 that Hence is a critical value of by the mountain pass theorem. Similar to the arguments above, problem admits another positive -symmetric solution with .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project is supported by the Natural Science Foundation of China (Grants nos. 11471235 and 11171247), and it is supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ130503).

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Copyright © 2014 Zhiying Deng and Yisheng Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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