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Positive symmetric solutions for a class of critical quasilinear elliptic problems in
Boundary Value Problems volume 2014, Article number: 154 (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.
In recent years, considerable attention has been paid to the following nonlinear elliptic problem with singular potential and critical Sobolev exponent:
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.
Recently, Deng and Jin  studied the existence of nontrivial solutions of the following singular semilinear elliptic problem:
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. .
where . Now we employ the following norm in :
By the Hardy inequality (2.2), we easily see that the above norm is equivalent to the usual norm .
The natural functional space to study problem (1.3) is the Banach space which is the subspace of consisting of all G-symmetric functions. In this paper we consider the following 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 N, p, μ, and s. From Kang , we see that satisfies the equations
for all . In particular, we have (let )
The function in (2.4) is the unique radial solution of the following limiting problem (see , Lemma 2.3]):
Moreover, the following asymptotic properties at the origin and infinity for and hold :
where , are positive constants and , are the zeros of the function
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.
Suppose that (q.1) and (q.2) hold. If
for some, where, then problem () has at least one positive solution in.
Suppose that (q.1) and (q.2) hold. Then we have the following statements.
Problem () has a positive solution if
and either (i) for someandsmall or (ii) for some constant, , small, and
Problem () admits at least one positive solution if exists and is positive,
and either (i) for someand largeor (ii) for some constants, , large, and
If on and
then problem () has at least one positive solution.
Suppose thatand. Then problem () has infinitely many G-symmetric solutions.
Letbe a constant. Suppose thatand (h.1) and (h.2) hold. If
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 ()
We associate with problem () a functional given by
By (q.1) and (2.1), we easily see that the functional . Now it is well known that there exists a one-to-one correspondence between the weak solutions of problem () and the critical points of ℱ. More precisely, the weak solutions of () are exactly the critical points of ℱ by the principle of symmetric criticality of Palais (see Lemma 3.1), namely satisfies () if and only if for all , there holds
Letbe a G-symmetric function; inimpliesin.
Letbe a weakly convergent sequence to u insuch that, , andin the sense of measures. Then there exists some at most countable set, , , , such that
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.
Suppose that (q.1) and (q.2) hold. Then thecondition inholds forif
The proof is similar to that in , Proposition 2]. We sketch the argument here for completeness. Let be a sequence for ℱ with . Then we easily deduce from (2.1) and (3.3) that is bounded in and we may assume that in . By Lemma 3.2 there exist measures ς, ν, and such that relations (a)-(e) of this lemma hold. Let be a singular point of measures ς and ν. As in paper , we define a function such that in , on and . By Lemma 3.1, , hence, using (2.1), the Hölder inequality, and the fact that , we get
Passing to the limit as , we deduce from (3.4) and Lemma 3.2 that
The above inequality says that the concentration of the measure ν cannot occur at points where , that is, if then . Combining (3.5) and (d) of Lemma 3.2 we find that either (i) or (ii) . For the point , similarly to the case , we have . This, combined with (e) of Lemma 3.2, implies that either (iii) or (iv) . To study the concentration of the sequence at infinity we need to consider the following quantities:
Obviously, and both exist and are finite. For , let be a regular function such that , for , for and . Then we deduce from the definition of that
We now claim that
In fact, using the elementary inequality for all and , we get
On the other hand, by the Hölder inequality and the Sobolev inequality, we have
Similarly, we have . The claim (3.7) is thereby proved. From (3.6) and (3.7), we derive . Moreover, since , we get . Therefore we conclude that either (v) or (vi) . We now rule out the cases (ii), (iv), and (vi). For every continuous nonnegative function ψ such that on , we obtain from (3.1) and (3.2) that
If (ii) occurs, then the set must be finite because the measure ν is bounded. Since functions are G-symmetric, the measure ν must be G-invariant. This means that if is a singular point of ν, so is for each , and the mass of ν concentrated at is the same for each . If we assume the existence of with such that (ii) holds, then we choose ψ with compact support so that for each and we obtain
a contradiction with (3.3). Similarly, if (iv) holds for , we choose ψ with compact support, so that , and we obtain
which contradicts (3.3). Finally, if (vi) holds we take to get
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
Let be the extremal function satisfying (2.4)-(2.9). We choose such that the assumption (2.10) holds. It is easy to check that there exist constants and such that for all . Simple arithmetic shows that there exists such that
We now choose such that and and set
where . From (2.10), (3.8), (3.9), and the definition of , we deduce that
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
First of all, we observe that due to the identity (2.6), inequality (2.10) is equivalent to for some , or equivalently
for some , where
Part (1), case (i). According to (3.10), we need to show that
for some . We choose so that for . This, combined with (2.8), implies that
as . On the other hand, for all , we obtain from (2.8) and the fact that
for some constant independent of ϵ. Combining (3.12) and (3.13), we get (3.11) for ϵ sufficiently small.
Part (1), case (ii). We choose so that for . Since , we deduce from (2.8) and the fact that
So by (2.8), (2.11), and the Lebesgue dominated convergence theorem we obtain
Thus (3.11) holds for ϵ sufficiently small.
Part (2), case (i). From (3.10) it is sufficient to show that
for some . We choose such that for all . This, combined with (2.7), implies that
as . Moreover, for all , we get from (2.7) and the fact that
for some constant independent of . These two estimates combined together give (3.14) for large.
Part (2), case (ii). We choose such that for all . Since , we have
Thus by (2.7), (2.12), and the Lebesgue dominated convergence theorem, we obtain
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]).
Let E be an infinite dimensional Banach space and letbe an even functional satisfyingcondition for each c and. Further, we suppose that:
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
Applying Lemma 3.4 with , we deduce from (q.1), (2.3), and (3.1) that
Since , there exists and such that for all u with . To find a suitable sequence of finite dimensional subspaces of , we set . Since the set Ω is G-symmetric, we can define , which is the subspace of G-symmetric functions of (see Section 2). By extending functions in by 0 outside Ω we can assume that . Let be an increasing sequence of subspaces of with for each m. Then there exists a constant such that
Consequently, if , with , then we write , with and . Therefore we obtain
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.
We follow the argument of , Lemma 2.2]. Let and such that . By (h.2), we can define the following integrals:
For sufficiently small, we deduce from (h.2), (2.1), (2.2), the Hölder inequality, and the fact , that
as . Also, for large enough, we obtain from (h.2), the Hölder inequality, and the fact , that
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. □
Since we are interested in positive G-symmetric solutions of (), we define a functional given by
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 .
Recall that the extremal function satisfies (2.4)-(2.9). By (h.2), we can choose such that and define a function such that , for , for and on . Using the methods in , , we deduce from (2.4)-(2.9) that
Set ; then by (4.4), (4.5), and (4.6) we have
Suppose that (h.1), (h.2), and (2.13) hold. Then there exists some, andon, such that
Recall that , which satisfies (4.7) and (4.8). In the following, we will show that satisfies (4.9) for sufficiently small. Set
with ; we deduce from and (h.2) that , for , and . Therefore can be achieved at some for which we obtain
Consequently, we deduce from (h.2), (4.7), and (4.10) that
On the other hand, the function attains its maximum at and is increasing on the interval , together with (4.3), (4.7), (4.11), and which is directly got from (h.2), we obtain
Furthermore, we easily check from (2.13) that
Consequently, choosing small enough, we deduce from (4.8), (4.12), and (4.13) that
Therefore we conclude that satisfies (4.9) for sufficiently small and the result follows. □
Suppose that (h.1) and (h.2) hold. Then thecondition inholds forif
Let be a sequence for with c satisfying (4.14). Then by (4.3) and the fact that , there exists such that for , we have
This implies that is bounded in . Consequently, just as in Lemma 3.3, we may assume that in and in ; moreover, in (see Lemma 4.1) and a.e. on . A standard argument shows that u is a critical point of , and hence
Now we set ; then we apply the Brezis-Lieb lemma  to the sequence and use the condition (h.2) and the fact that u is a critical point of to obtain
Consequently, for a subsequence of one gets
It follows from (2.3) that , which implies either or . If , we deduce from (4.15), (4.16), and (4.17) that
which contradicts (4.14). Therefore, we obtain as , and hence, in . The proof of this lemma is completed. □
Proof of Theorem 2.3
For any , we obtain from (h.2), (2.1), (2.3), (4.3), and the Hölder inequality
Therefore, there exist constants and such that for all . Moreover, since as , there exists such that and . Now we set
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 (). □
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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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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