Sign Changing Solutions for Coupled Critical Elliptic Equations

Wang, Xin; Yue, Xiaorui

doi:https://doi.org/10.1155/2020/9841640

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9841640 | https://doi.org/10.1155/2020/9841640

Sign Changing Solutions for Coupled Critical Elliptic Equations

Xin Wang¹and Xiaorui Yue²

Academic Editor: Guozhen Lu

Received01 Dec 2019

Accepted03 Jan 2020

Published11 Feb 2020

Abstract

In this paper, we consider the coupled elliptic system with a Sobolev critical exponent. We show the existence of a sign changing solution for problem for the coupling parameter . We also construct multiple sign changing solutions for the symmetric case.

1. Introduction

In this paper, we consider the following coupled elliptic system with a Sobolev critical exponent:where is a bounded smooth domain, , , for , and is the first eigenvalue of with the Dirichlet boundary condition.

In recent years, the following coupled elliptic system has attracted much interest:

is a bounded smooth domain or , . System (2) arises when we consider the standing wave solutions of the two coupled Gross–Pitaevskii equations:

The system above has applications in many physical problems (see [1, 2]). It also arises in the Hartree–Fock theory for a double condensate, which is a binary mixture of Bose–Einstein condensates in two different hyperfine states and (see [3] and the references therein). Considering the solitary wave solutions of system (3), we set for . Then, it is reduced to system (2). For the subcritical case, i.e., , the existence of least energy and other finite energy solutions as well as the existence and multiplicity of positive and sign-changing solutions are studied in [2, 4–23] and the references therein. For the critical case, i.e., , the existence of a positive least energy solution is proved when β is negative, positive small, and positive large in [24]. For the higher dimension , the authors in [25, 26] also consider the following critical case:

Note that when , system (4) is the same as system (2). But interestingly, the authors in [25] find different results for the higher dimension case from that of . The authors in [27] also consider the even case:

They show that when , system (5) has a sign changing solution for any .

Recall that when , system (4) becomes the famous Brezis–Nirenberg problem:which are two different independent scalar equations. In [28], the authors show that when , and , equation (6) has a ground state solution. For the even case,

The authors in [29] show that when , and , equation (7) has sign changing solutions for . In [30, 31], when , the authors obtained nontrivial solutions of (7) for .

In fact, in the pioneering paper [28] of Brezis and Nirenberg, they also study a more general equation including the following classical case:where . From [28], we know that when , then problem (8) has positive ground state solutions with the energy:where S is the sharp imbedding constant from into .

Based on these, firstly, we try to show similar results in [29–31] for the even case of the general equation (8), i.e.,

Then, naturally and interestingly, we guess that the more general system corresponding to system (5) also has similar result of the existence of sign changing solutions. That is, we consider the more general critical elliptic system as follows:

Precisely, we get the following results.

Theorem 1. Assume that , for ; then, (10) has sign changing solutions for and the energy is the least one among all sign changing solutions.
Then, we can consider the case . The following equation is one case of (10):where . Let and be the sign changing solutions of (12) when , and , respectively (the latter case really exists and we shall show it later). By Lemma 1 in [25], the system below has a positive solution:Then, we can construct solutions of (11) by using as in [2, 24, 25].

Theorem 2. Let be a solution of (13). Assume that and one of the following happens:(1)(2)Then are sign changing solutions of system (11).

We also get a least energy semi-sign changing solution for system (11).

Theorem 3. Assume that , and . Then, system (11) has a semi-sign changing solution with one component changing sign and the other one positive, and the energy is the least one among all these kinds of solutions.

Next, we give the proof of Theorem 1 in Section 2. And we shall show Theorems 2 and 3 in Sections 3 and 4, respectively.

2. Proof of Theorem 1

In this section, we consider scalar case (10). Fixing or 2, we show the existence result. The proof is similar to that in [29]. Considering we need this result in the following two Theorems 2 and 3, we verify it also for the completeness of the current paper. The working space and some notations shall be given firstly. We assume that all the integrations below are taken over if without special specification. Since , we can define the equivalent inner product in bywhich gives rise to a norm denoted by . We also use and for convenience. Then, the energy functional of equation (10) is

Recall is a ground state solution of (8), that is,

For the sign changing case, we define the manifold as follows:where and the condition in the definition of (17) is that

It is easy to check that Then, we define

We need a conclusion in [32].

Lemma 1 (see [32]). Consider a rectangle and a continuous function If hold for all , then has a zero inside .

Similarly as in [29], we set

Then, (18) is equivalent to . Define

Since , it is easy to see that . Then, we have the following lemma.

Lemma 2. .

Proof. For any , , we haveThen,Denote the map satisfyingBy (23), we haveOn the other hand, for any , we haveThen, by Lemma 1, we have that there exists such that , i.e., . Consequently,This completes the proof.
Now, we have an upper estimate for .

Lemma 3. .

Proof. Since and on , there exists for some near and small to be decided later. Take a cutoff function with and for . LetThen (see [33, 34]), is the solution of the following equation:Set ; then (see [35]),Now, we show thatIn fact, by (30) and since , we haveThen by Lemma 2, the conclusion follows.□
SetThen, we show that satisfies a local condition in the following sense.

Lemma 4. Under the assumptions in Theorem 1, for any , if satisfies thatthen we have that has a convergent subsequence in .

Proof. It is easy to check that is bounded in , so there exists such that weakly in up to a subsequence and strongly in for . Then, and weakly in . By the Sobolev inequality, we have that there exist positive constants independent of n such that and thenSincewe have either strongly in or . Considering the latter case, we haveTherefore,is a contradiction. That is, we must have strongly in and .
Now we are ready to show Theorem 1 based on a deformation lemma (see [36–38] for instance).

Proof of Theorem 1. Obviously, is coercive on and then there exists a minimizing sequence . Let such thatThen, by Lemma 2, we haveBy the well-known deformation lemma, we have that there exists a sequence such thatObviously, is bounded and there exists such that weakly in ; then, by the definition of weak convergence.
Now, we claim that . In fact, by (41), there exists a sequence withThen, for n large enough depending on ,(44), (45), and (43) implies that . Since , the claim comes true.
Therefore, by Lemmas 3 and 4, we have that strongly in with , i.e., is a sign changing solution of equation (10) and the energy is the least one among all sign changing solutions of (10).

3. Proof of Theorem 2

Firstly, inspired by [30, 31], we show the following theorem about the scalar equation by the minimax method (see [39]), and we would like to just give the sketch for the completeness of the current paper. We also use the same notations as in Section 2 whenever no confusion arises.

Denote … as the normalized eigenfunctions of , corresponding to positive eigenvalues , counted with their multiplicity. We set as the k dimensional space in . For the fixed , define

We may suppose that for some . Especially, we set .

Theorem 4. Assume that , for , then equation (10) has a nontrivial solution. Moreover, when , the solution changes sign.

Proof. It is easy to check that satisfies the following geometry.(a)For , there exist such that with and for (b)For any finite dimensional subspace , it holds that is bounded in , where As in [36], we define the minimax value:whereObviously, . We also set another minimax value:whereand γ is the Krasnoselskii genus. It can be verified that . By properties (a) and (b), we can show thatand satisfies the condition. Also by a deformation lemma, there exists a sequence of and then in . That is, u is a nontrivial solution of equation (10).
Multiply the equations in (10) with the first eigenfunction and integrate over ; we haveThen, it is easy to see that when any solution of (10) must change sign.

Proof of Theorem 2. The proof is by direct computation. Recall that are sign changing solutions of (12) being well defined. Since satisfy (13), for any , we haveThat is, satisfy the first equation in system (11). Similarly, we can verify that Theorem 2 holds.

4. Proof of Theorem 3

Inspired by [27, 29], we now consider the general case of system (11), i.e., we do not assume that . We always assume that for . Firstly, we introduce the product space as the working space. Define the inner product aswhich gives rise to a norm on denoted by . Recall that solutions of (11) are critical points of the following energy functional:

For with , we denote

Similar to related to the scalar equation, we defineThat is,

Let with and , it is easy to see that there exist such that . Thus, . By Sobolev inequality, it is easy to see that for any , there exists a constant such that

Set

Then, we define

We firstly have a lower bound for .

Lemma 5. Assume that ; then I is coercive on and .

Proof. We may assume that . For , note that for , it holds thatThen, we have(63) implies that I is coercive on . Then, by (59), we have .□
We shall give an upper bound for later by defining another manifold and the infimum on it. The idea is similar to that in [27], but since the corresponding equation (10) is different and more complex as well as system (11), some new tricks should be used in the current paper, so we shall give the details of the proofs. LetObviously, . Then, we defineBefore the estimate for the bound of , we need some preliminaries.

Lemma 6. Assume that ; then, for any , is attained only by . Similarly, for any , is attained only by .

Proof. For ; setThen, andSetDenoteThen,Note that . We can show that is negative definite. Thus, is the unique maximum point of .

Lemma 7. Assume that ; then, for any with , we have that there exist unique such that . Moreover, are continuous with respect to .

Proof. Proof. For any with , we denoteThen, and . for some is equivalent toBy (75), we havewhere implies that . Since and for small enough, , there exists a unique such that ; thus, . Combining (76) and (77), it is left to show thathas a solution . Note that and ; by direct calculation, we can check thatThis implies that (75) and (76) have a positive solution . The uniqueness of follows from Lemma 6.
For the continuity of with respect to , we take a sequence with strongly in . We may assume that by replacing by if necessary. That is, . Using similar denotations to (74) and rewriting for convenience, we haveIf , then by (81), we have . Therefore,is a contradiction. Thus, are uniformly bounded and we may assume that as . By (80) and (81), we have . Then, , that is, .
Now, we show the following:

Lemma 8. Assume that , then

Proof. (i) We first show that . In [27], the authors show that (11) has a positive least energy solution; we denote it as with the energy and . For small, take such that and Let with and on . Set . Then, there exists a positive constant such that . Define . Then, for we have By Lemma 7, there exist such that . That is, Then similar to the proof of (83), it can be shown that are uniformly bounded. Thus, up to a subsequence, there exist such that Let in (88) and (89); it implies that . Then, , and we may assume that for small. By (87), we have Let be the solution of problem (8) when ; then, and from [40], it holds thatThus,(ii)Next, we show that Let be the sign changing solution of (8) with for . Then, and For any , we may take with Then, and Recall ϕ and in the above step ; we define . Since , we haveThen, there exists such thatand It is easy to see that , where is the positive solution of problem (8) in the ball for . Then by (91), we haveFor , we define . Let and recall that is defined in Lemma 7; then, we can define a continuous map byThen, we have the following conclusion and recall that is defined in (60).

Lemma 9. For any with , there exists a small such that

Proof. It is easy to see that there exist constants such thatSince and , we haveSimilarly, note that ; we haveSince , we have . Then,Similarly, we haveSince are continuous with respect to ρ, (101), (102), (103), and (104) imply (97), (98), and (99), respectively.□
Now, we show the existence of a sequence of I. Precisely, we have the following important lemma.

Lemma 10. There exist a sequence and a constant such that

Proof. By Lemma 5 and the Ekeland’s variational principle, there exists a minimizing sequence such thatLet ; by Lemma 9, there exists such thatWe claim that there exists a sequence such thatIf not, then we assume that there exists small such thatwhereand is the neighborhood of S. Then, by the well-known deformation lemma, there exist a continuous map and an such that(1)(2)(3)By Lemma 6, for n large enough, we have thatThen, by the properties (2) and (3) of ξ and (109), we haveSimilarly as in (20), we defineWe rewrite for convenience. implies that By (107) and (108), we have thatThen, by the continuity of E, we can defineBy definition of , there exists such that for any . Then, for any . Thus, . Hence, . That is, . But by (115), we have , a contradiction with the definition of . Therefore, the claim (111) becomes true.
Now, we can choose such that Since , we have . Recall that and ; then by Lemma 6, we have . By (59), we have that there exists a constant independent of n such that
SetBy (111), there exist two Lagrange multipliers such thatThen, and ; i.e.,Similarly as in (74), we denoteNote that , we haveConsequently, by (59), we haveSetThen, by (122) and (123), we haveNote thatWe haveThus, for some constant . By (128), we have . Without loss of generality, we may assume that ; then by (123) and (59), we have . Then, (121) implies that . Thus, we complete the proof.
We also need an important lemma which is proved in [25].

Lemma 11 (see [25]). Assume that weakly in as ; then, passing to a subsequence, it holds that

Proof of Theorem 3. Proof of Theorem 3. Let be the sequence obtained in Lemma 10. Then, up to a subsequence, there exists such that weakly in . Then, andSet . By Lemma 11, (133), and (134), we haveThen,

Case 1. 1. Assume that . Then, in and for . From (135) and (136), we haveThen,Thus, by (137), we havea contradiction with Lemma 8, (85), and (9). Therefore, Case 1 is impossible.

Case 2. Assume that . Then, (139) still holds and which implies that is a nontrivial solution of (10) with . Thus, . By (137), we havea contradiction with Lemma 8 and (85). Therefore, Case 2 is impossible.

Case 3. Assume that . Then, (140) still holds and which implies that is a nontrivial solution of (10) with . Thus, .(i)If strongly in , then is a sign changing solution of (10) with . So . By (137), we have a contradiction with Lemma 8.(ii)If strongly in , then by (132) and Brezis–Lieb lemma, we can also show thatBy (137), we havea contradiction with Lemma 8 and (85).
Therefore, Case 3 is impossible.
Now, we have shown that and then is a nontrivial solution of (11). If strongly in , then (144) holds anda contradiction with Lemma 8. Thus, strongly in and then is sign changing, . On the other hand, by (137), we haveThen, we have and strongly in . Then and . Similarly to the proof in Lemma 10, we have . By the maximum principle, we have in . Consequently, is a least energy sign changing solution of system (11) with sign changing and positive. Thus, we complete the proof.

5. Conclusion

In this paper, we study a coupled nonlinear Schrödinger system with critical exponents which arise in many physical problems. By the modified Nehari manifold method and some mathematical skills for estimations of the energy, we show the existence of the least energy sign changing solution for the general system. Besides, we construct multiple solutions of the system for the symmetrical case.

Data Availability

No data were used to support this study and all materials are available in this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors studied this problem and read and approved the final manuscript.

Acknowledgments

This work was supported by the NSFC (11601109), Hainan Association of Science and Technology plans for Youth Innovation (201503), and Natural Science Fund of Hainan Province (20161001).

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Copyright © 2020 Xin Wang and Xiaorui Yue. 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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