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Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential

Tsing-San Hsu

Author Affiliations

Department of Natural Sciences in the Center for General Education, Chang Gung University, Kwei-San, Tao-Yuan, 333, Taiwan, ROC

Boundary Value Problems 2012, 2012:116  doi:10.1186/1687-2770-2012-116

 Received: 12 March 2012 Accepted: 3 October 2012 Published: 17 October 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider singular elliptic systems involving a strongly coupled critical potential and concave nonlinearities. By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.

MSC: 35J60, 35B33.

Keywords:
Palais-Smale condition; Nehari manifold; strongly coupled; elliptic system; critical potential

1 Introduction and main results

In this paper, we consider the following elliptic system:

(1.1)

where is a smooth bounded domain such that , , is the critical Sobolev exponent, is the best Hardy constant and denotes the completion of with respect to the norm and is defined as the completion of the with respect to the norm defined by for .

Definitions of strongly and weakly coupled terms are as follows.

The terms and () are weakly coupled, () is strongly coupled when L or K is a derivative operator. Thus, is strongly coupled when and are positive.

The parameters in (1.1) satisfy the following assumption.

(ℋ) , , , , , , , , .

The corresponding energy functional of (1.1) is defined in by

where and . Then and the duality product between and its dual space is defined as

where and denotes the Fréchet derivative of J at . A pair of functions is said to be a weak solution of (1.1) if

Therefore, a weak solution of (1.1) is equivalent to a nonzero critical point of [1].

Problem (1.1) is related to the well-known Hardy inequality [2]

(1.2)

If , by (1.2), is an equivalent norm of H, the operator L is positive and the first eigenvalue of L and the following best constant are well defined:

(1.3)

where is the completion of with respect to . Note that is the well-known best Sobolev constant. For , the constant is achieved by the following extremal functions [3]:

(1.4)

where is a radially symmetric function

On the other hand, for any , , , and , , by the Young and Sobolev inequalities, the following best constants are well defined on the space :

(1.5)

We define

(1.6)

Since f is a continuous function on such that . Then there exists such that

(1.7)

Set , , and . Then (1.1) reduces to the semilinear scalar problems that have been extensively investigated by many authors. See [4-6] and the references therein.

Regular semilinear elliptic systems have been studied extensively and many conclusions have been established. For example, Alves et al. studied in [7] an elliptic system and some important conclusions had been obtained. However, the elliptic systems involving the Hardy inequality have seldom been studied and we only find some results in [8-16]. Thus it is necessary for us to investigate the related singular systems deeply. Among the references above, the elliptic systems involving the Hardy inequality and concave-convex nonlinearities had been studied only in [12]. In this paper, only the case of (1.1) involving multiple strongly-coupled critical terms is considered.

Let be the Lebesgue measure of Ω. We define the following constant:

(1.8)

Then the main results of this paper can be concluded in the following theorems and the conclusions are new to the best of our knowledge. It can be verified that the intervals in Theorems 1.1 and 1.2 for the parameters , , μ and q are allowable.

Theorem 1.1Suppose that (ℋ) holds and. Then problem (1.1) has at least one positive solution.

Theorem 1.2Suppose that (ℋ) holds, , and. Then there existssuch that problem (1.1) has at least two positive solutions for allandsatisfying.

This paper is organized as follows. Some preliminary results and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1 and 1.2 are proved in Section 4.

2 The local Palais-Smale condition

Throughout this paper, we always assume that the assumption (ℋ) holds, denotes the norm of the space H, by the Hardy inequality is equivalent to , i.e.,

denotes the first eigenvalue of the operator L, means the norm of the space , is the dual space of E. for all and . is said to be nonnegative in Ω if and in Ω. is said to be positive in Ω if and in Ω. is a ball in . denotes a quantity satisfying , means as and is a generic infinitesimal value. In particular, the quantity means that there exist the constants such that as ε is small. We always denote positive constants as C and omit dx in integrals for convenience.

Lemma 2.1Ifis a (PS)c-sequence ofJwithinE, thenand, where

Proof Let and . Since is a (PS)c-sequence of J with in E, we can deduce that , and therefore , that is,

Consequently,

From the Hölder inequality it follows that

Thus, the proof is complete. □

Lemma 2.2Ifis a (PS)c-sequence of the functionalJ, thenis bounded inE.

Proof See Hsu [[12], Lemma 2.2]. □

Lemma 2.3Suppose that (ℋ) holds. ThenJsatisfies the (PS)ccondition for all, where

(2.1)

Proof We follow the argument in [15]. Let be a (PS)c-sequence of J with . Write . We know from Lemma 2.2 that is bounded in E, and then up to a subsequence, z is a critical point of J. Furthermore, we may assume that , weakly in H and , strongly in for all and , a.e. in Ω. Hence, we have that

(2.2)

Set , and . From the Brézis-Lieb lemma [17] it follows that

(2.3)

and by Lemma 2.1 in [18] we have

(2.4)

Since , and by (2.2) to (2.4), we can deduce that

(2.5)

and

Hence, we may assume that

(2.6)

If , the proof is complete. Assume ; then from (2.6) and the definition of it follows that

which implies that

(2.7)

In addition, from (2.5) to (2.7) and Lemma 2.1, we get

which is a contradiction. Therefore, the proof of Lemma 2.3 is complete. □

3 Nehari manifold

Since J is unbounded below on E, we need to consider J on the Nehari manifold

Thus, if and only if

(3.1)

By the Hölder inequality and the definition of it follows that

(3.2)

Lemma 3.1The functionalJis coercive and bounded below on.

Proof Suppose that . From (3.1) and (3.2) we get

(3.3)

(3.4)

Thus, J is coercive and bounded below on . □

Define . Then for all we have

(3.5)

(3.6)

We split into three parts:

Lemma 3.2Suppose thatis a local minimizer ofJonand. Thenin.

Proof The proof is similar to that of [19] and the details are omitted. □

Lemma 3.3for all.

Proof We argue by contradiction. Suppose that there exist such that and . Then the fact together with (3.5) and (3.6) imply that

(3.7)

and

(3.8)

By (1.5) and (3.7) we have

which implies that

(3.9)

By (3.2) and (3.8) we have

(3.10)

From (3.9) and (3.10) it follows that

By Lemma 3.3, we write and define

Lemma 3.4

(i) for all.

(ii) There exists a positive constantdepending on, , q, N, , andsuch thatfor all.

Proof (i) Let . By (3.1) and (3.6) it follows that

(3.11)

According to (3.1) and (3.11), we have that

which implies that .

(ii) Suppose that and . By (1.7), (3.1) and (3.5) we have that

which implies that

(3.12)

From (3.4) and (3.12) it follows that

where is a positive constant. □

Lemma 3.5Suppose thatandwith. Then there exist uniquesuch thatand. In particular, we have

and.

Proof The proof is similar to that of [20] and is omitted. □

For each with , we write

Then we have the following lemma.

Lemma 3.6Suppose thatandwith. Then there exist uniquesuch that, and

Proof The proof is almost the same as that in [[20], Lemma 2.7] and is omitted here. □

4 Proof of Theorems 1.1 and 1.2

Lemma 4.1

(i) If, then the functionalJhas a (PS)-sequence.

(ii) If, then the functionalJhas a-sequence.

Proof The proof is similar to that of [21] and is omitted. □

Lemma 4.2Suppose that. ThenJhas a minimizersuch thatis a positive solution of (1.1) and.

Proof By Lemma 4.1(i), there exists a (PS)-sequence of J such that

(4.1)

Since J is coercive on (see Lemma 3.1), we get that is bounded in E. Passing to a subsequence (still denoted by ), we can assume that there exists such that

(4.2)

which implies that

(4.3)

First, we claim that is a solution of (1.1). By (4.1) and (4.2), it is easy to see that is a solution of (1.1). Furthermore, from and (3.3), we deduce that

(4.4)

Taking in (4.4), by (4.1), (4.2) and the fact , we get

Therefore, is a nontrivial solution of (1.1).

Next, we prove that strongly in E and . Noting and applying the Fatou lemma, we have

Therefore, and . Set . By the Brézis-Lieb lemma [17], we get

Then standard argument shows that strongly in E. Moreover, we have . Otherwise, if , then by Lemma 3.5 there exist unique such that and . Since

there exists such that . By Lemma 3.5 we get that

which is a contradiction. Since and , by Lemma 3.2 we may assume that is a nontrivial nonnegative solution of (1.1).

In particular , . Indeed, without loss of generality, we may assume that . Then as is a nontrivial nonnegative solution of

by the standard regularity theory, we have in Ω and

Moreover, we may choose such that

Now,

and so by Lemma 3.6 there is unique such that . Moreover,

and

This implies

Finally, from the maximum principle [22] we deduce that in Ω and is thus a positive solution of (1.1). □

Let be defined as in (1.4) and set , where is a cut-off function:

The following results are already known.

Lemma 4.3[4]

Aswe have the following estimates:

(4.5)

(4.6)

(4.7)

Lemma 4.4[11]

Suppose that (ℋ) holds, is defined as in (1.6) andare the minimizers ofdefined as in (1.4). Thenand has the minimizers, where.

Lemma 4.5Under the assumptions of Theorem 1.2, there existandsuch that for allthere holds

(4.8)

In particular, for all.

Proof For all , define the functions and

Note that and as t is closed to 0. Thus, is attained at some finite with . Furthermore, , where and are the positive constants independent of ε.

Choose small enough such that for all . Set . Then for all and , which implies that there exists satisfying , for all . Note that

(4.9)

From (4.9) and Lemmas 4.3, 4.4 it follows that

Consequently,

and

where we have used the assumption .

Therefore we can choose , such that

The definition of in Lemma 2.1 implies that

Note that

Taking ε small enough, there exists such that for all ,

(4.10)

Choose . Then for all there holds

(4.11)

Finally, we prove that for all . Recall that . By Lemma 3.5, the definition of and (4.11), we can deduce that there exists such that and

The proof is thus complete by taking . □

Lemma 4.6Set. Then for all, problem (1.1) has a positive solutionsuch thatand.

Proof By Lemma 4.1, there exists a -sequence of J for all . From Lemmas 2.3, 3.4 and 4.5, it follows that and J satisfies the condition for all . Since J is coercive on , we get that is bounded in E. Therefore, there exist a subsequence (still denoted by ) and such that strongly in E and for all . Since and , by Lemma 3.2 we may assume that is a nontrivial nonnegative solution of (1.1). Moreover, by (3.7) and , we get

This implies that and . From the strong maximum principle [22] it follows that is a positive solution of (1.1). □

Proof of Theorems 1.1 and 1.2. By Lemma 4.2, we obtain that (1.1) has a positive solution for all . On the other hand, from Lemma 4.6, we can get the second positive solution for all . Since , this implies that and are distinct. □

Competing interests

The author declares that he has no competing interests.

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