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:
PalaisSmale condition; Nehari manifold; strongly coupled; elliptic system; critical potential1 Introduction and main results
In this paper, we consider the following elliptic system:
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.
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 wellknown Hardy inequality [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:
where is the completion of with respect to . Note that is the wellknown best Sobolev constant. For , the constant is achieved by the following extremal functions [3]:
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 :
We define
Since f is a continuous function on such that . Then there exists such that
Set , , and . Then (1.1) reduces to the semilinear scalar problems that have been extensively investigated by many authors. See [46] 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 [816]. 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 concaveconvex nonlinearities had been studied only in [12]. In this paper, only the case of (1.1) involving multiple stronglycoupled critical terms is considered.
Let be the Lebesgue measure of Ω. We define the following constant:
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 PalaisSmale 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)_{c}condition for all, where
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
Set , and . From the BrézisLieb lemma [17] it follows that
and by Lemma 2.1 in [18] we have
Since , and by (2.2) to (2.4), we can deduce that
and
Hence, we may assume that
If , the proof is complete. Assume ; then from (2.6) and the definition of it follows that
which implies that
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
By the Hölder inequality and the definition of it follows that
Lemma 3.1The functionalJis coercive and bounded below on.
Proof Suppose that . From (3.1) and (3.2) we get
Thus, J is coercive and bounded below on . □
Lemma 3.2Suppose thatis a local minimizer ofJonand. Thenin.
Proof The proof is similar to that of [19] and the details are omitted. □
Proof We argue by contradiction. Suppose that there exist such that and . Then the fact together with (3.5) and (3.6) imply that
and
By (1.5) and (3.7) we have
which implies that
By (3.2) and (3.8) we have
From (3.9) and (3.10) it follows that
which is a contradiction. □
By Lemma 3.3, we write and define
Lemma 3.4
(ii) There exists a positive constantdepending on, , q, N, , andsuch thatfor all.
Proof (i) Let . By (3.1) and (3.6) it follows that
According to (3.1) and (3.11), we have that
(ii) Suppose that and . By (1.7), (3.1) and (3.5) we have that
which implies that
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
Proof The proof is similar to that of [20] and is omitted. □
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 asequence.
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
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
which implies that
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
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 BrézisLieb 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
which is a contradiction.
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 cutoff function:
The following results are already known.
Lemma 4.3[4]
Aswe have the following estimates:
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
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
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 ,
Choose . Then for all there holds
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.
Acknowledgements
The author was grateful for the referee’s helpful suggestions and comments.
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