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Multiplicity of Solutions for an Elliptic Problem with Critical Sobolev-Hardy Exponents and Concave-Convex Nonlinearities
We study the existence of multiple solutions for the following elliptic problem: We prove that if , then there is a , such that for any , the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result (Azorero and Alonso, 1991).
1. Introduction and Main Results
In this paper, we study the existence of multiple solutions to the following elliptic problem: where is a smooth bounded domain containing the origin , is the p-Laplacian of , , , , , , and is the critical Sobolev-Hardy exponent; note that is the critical Sobolev exponent.
In this paper, we use to denote the usual weighted space with the weight . In , for , we use the norm By (3), this norm is equivalent to the usual norm . By the Hardy inequality and the Sobolev-Hardy inequality, for , , we can define the following best constants: Note that is the best constant in the Sobolev inequality, that is, The energy functional of (1) is defined as follows: Then, is well defined on and belongs to . The solutions of problem (1) are then the critical points of the functional .
In recent years, the quasilinear problems related to Hardy inequality and Sobolev-Hardy inequality have been studied by some authors [3–7]. Ghoussoub and Yuan  studied problem (1) with , , and and proved the existence results of positive solutions and sign-changing solutions. Kang in [3, 4] studied (1,1) when and verified the existence of positive solutions of (1) when the parameters , , , , satisfy suitable conditions. To the best of our knowledge, there are few results of problem (1) involving the p-sublinear of . We are only aware of the works [6–9] which studied the existence and multiplicity of solution of problem (1) involving weight functions. Azorero and Alonso  studied problem (1) with , and proved that there exists , such that (1) has infinitely many solutions for . Hsu  studied problem (1) and proved that there exists such that (1) has at least two positive solutions for . In this paper, we study (1) and extend the results of [7, 9].
Throughout this paper, let be the positive constant such that , where . By Holder inequalities, for all , we obtain where is the volume of the unit ball in . The following inequality comes from the paper :
Now we are ready to state our main results.
Theorem 1. If is a bounded domain in , and , then there is a such that problem (1) possesses infinitely many weak solutions in for any .
2. The Palais-Smale Condition
Let be a Banach space and be the dual space of . The functional is said to satisfy the Palais-Smale condition at level (), if any sequence satisfying contains a subsequence converging in to a critical point of the functional . In this paper, we will take .
Lemma 2. Let be a Palais-Smale sequence for defined by (7), that is, If and , and depends on , , , then, there exists a subsequence , strongly convergent in .
Proof. By (11) and (12), it is easy to prove that the sequence is bounded in . Passing to a subsequence if necessary, we may assume that, as ,
Then, is a solution of problem (1). By the concentration compactness principle (see [10, 11]), there exists a subsequence, still denoted by , at the most countable set , a set of different points , sets of nonnegative real numbers , , and nonnegative real numbers and , such that
where is the Dirac mass at .
Case 1 ( and ). We claim that is finite, and, for any , either
In fact, let be small enough such that and for , . We consider , such that It is clear that the sequence is bounded in . Note that By (13), (16), and the Holder inequality, we obtain From (12)(18), we get that By the Sobolev inequality, , hence, we deduce that which implies that is finite.
Now we consider the possibility of concentration at the origin. Let be small enough such that , . Take such that By (13) and (14), we also get that By the definition of , we deduce that From (22) we have which implies that or We will prove that (25) and are not possible. By (13) and (14), By applying the Holder inequality at (26), we have Let , , . This function obtains its absolute minimum (for ) at point . That is, where because of . But this result contradicts the hypothesis. Then, and we conclude.
Case 2 (, then ). We only need to consider the possibility of concentration at the origin. Let be small enough such that . Take a smooth cut-off function centered at the origin such that , for , for , and . By (13) and (14), we get that By the definition of , we deduce that From (30), we have which implies that or We will prove (33) is not possible. From the above arguments and (8), we conclude that Let , , . This function obtains its absolute minimum at point . That is, But this result contradicts the hypothesis. Hence, up to a subsequence, we obtain that strongly in .
Thus, the proof of the Lemma is completed.
3. Existence of Infinitely Many Solutions
In this section, we will prove our main result of Theorem 1. We first recall some concepts and results in minimax theory.
Let be a Banach space, and denote all closed subsets of which are symmetric with respect to the origin. For , we define the genus by if the minimum exists, and if such a minimum does not exist, then we define . The main properties of the genus are contained in the following lemma (see  for the details).
Lemma 3. Let . Then one has the following.(1)If there exists , odd, then .(2)If , then .(3)If there exists an odd homeomorphism between and , then .(4)If is the sphere in , then .(5)Consider .(6)If , then .(7)If is compact, then , and there exists such that , where .(8)If is a subspace of with codimension , and , then .
Let be a Banach space and be a functional on . Denote , .
Given the functional , under the hypothesis , using Sobolev’s equality and (9), we obtain If we define for then Because and , as , it is easy to see that there exists such that, if , attains its positive maximum.
From the structure of , we see that there are constants , such that , if , if , and if . Following , let be nonincreasing, such that and let ; we consider the truncated functional Similar to (39), we have , where Clearly, for and if , , if , and if , is strictly increasing, and so , if . Consequently, for .
Lemma 4. Consider .
If , then and for all in a small enough neighborhood of .
There exists , such that if , then verifies a local Palais-Smale condition for .
Proof. (1) and (2) are immediate. To prove (3), observe that all Palais-Smale sequences for with must be bounded; then, by Lemma 2, if verifies , there exists a convergent subsequence.
Now, we use the idea in  to construct negative critical values of via genus.
Lemma 5. Given , there is an , such that
Proof. Fix ; let be an -dimensional subspace of ; we take with norm for ; we have
Since is a space of finite dimension, all the norms in are equivalent. If we define
and we can choose (which depends on ), and , such that if and .
Let . Consider ; therefore, by Lemma 3, we see that We are now in a position to prove our main result.
Proof of Theorem 1. Let , , , and suppose that where is the constant given by Lemma 4. We claim that if are such that , then .
In fact, denote ; by Lemma 5, we see that for any , there is a , such that . Since is continuous and even, and . As is bounded from below, we see that for all .
Suppose that ; then satisfies condition by Lemma 2, and it is easy to see that is a compact set.
If , then there is a closed and symmetric set with and by Lemma 3. Since , we can also assume that the closed set . Since satisfies condition for , by the Deformation Lemma, there is an odd homeomorphism, such that for some with .
Since , there exists an , such that
But by Lemma 3 and , we have Hence, and , which contradicts to (50). So we have proved that .
Now if for all , we have , . If all are distinct, then , and we see that is a sequence of distinct critical values of ; if for some , there is a such that then which shows that contains infinitely many distinct elements.
Since if , we see that there are infinitely many critical points of . The theorem is proved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by Ningbo Scientific Research Foundation (2009B21003), K. C. Wong Magna Fund in Ningbo University, NSF of Hebei Province (A2013209278), and National Natural Science Foundation of China (nos. 61271398, 11220248 and 11175092).
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