Multiple Positive Solutions and Estimates of Extremal Values for a Nonlocal Problem with Critical Sobolev Exponent and Concave-Convex Nonlinearities

Shi, Zhigao; Qian, Xiaotao

doi:https://doi.org/10.1155/2022/1011342

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 1011342 | https://doi.org/10.1155/2022/1011342

Multiple Positive Solutions and Estimates of Extremal Values for a Nonlocal Problem with Critical Sobolev Exponent and Concave-Convex Nonlinearities

Zhigao Shi¹and Xiaotao Qian²

Academic Editor: Simone Secchi

Received17 Mar 2022

Revised03 May 2022

Accepted21 May 2022

Published02 Jun 2022

Abstract

We are concerned with the following nonlocal problem involving critical Sobolev exponent where is a smooth bounded domain in , , , , and are positive parameters. We prove the existence of two positive solutions and obtain uniform estimates of extremal values for the problem. Moreover, the blow-up and the asymptotic behavior of these solutions are also discussed when and . In the proofs, we apply variational methods.

1. Introduction and Main Results

In this paper, we study a new class of Kirchhoff type problem with critical exponent and concave-convex nonlinearities where is a smooth bounded domain in ( is the critical exponent in dimension four), , , , and are positive parameters.

We call a Kirchhoff type problem since the presence of the term , which means that is no longer a pointwise identity. Such nonlocal problem arises in various models concerning physical and biological systems, see, e.g., [1–3]. Among others, Kirchhoff [2] built a model defined by the equation where represents the lateral displacement, denotes the mass density, is the initial tension, denotes the area of the cross-section, denotes the Young modulus of the material, and is the length of the string. This equation is an extension of the classical D’Alembert wave equation for free vibrations of elastic strings.

Different from the traditional Kirchhoff type problem, the sign of nonlocal term included in is negative, which causes some interesting difficulties. In the past few years, much attention has been paid to the existence, multiplicity, and the behaviour of solutions for this kind of nonlocal problem but without critical growth. In particular, Yin and Liu [4] were concerned with the following problem where and is a bounded domain in with and succeeded to find the problem (3) admits at least two nontrivial solutions. In [5, 6], sign-changing solutions to (3) were further obtained. When and the nonlinear term has an indefinite potential, Lei et al. [7] and Qian and Chao [8] established the existence of positive solution of (3) for and , respectively. For the singular nonlinearity, two positive solutions to (3) with were proved in [9]. In our previous work [10], we obtained two positive solutions of with , as well as their blow-up and asymptotic behavior when . For more related results, we refer the interested readers to [11–15] and the references therein.

In 1994, Ambrosetti et al. [16] first studied the following critical local problem involving concave-convex nonlinearities where and is a smooth bounded domain. The authors proved that there exists such that the problem (4) has two positive solutions for and no positive solutions for . Since then, many scholars have considered problems with critical exponent and concave-convex nonlinearities, see, e.g., [7, 16–21]. Also, the problem (4) of traditional Kirchhoff type is studied in [22–26] and the reference therein. An interesting question now is whether the same existence results as in [16] occur to the nonlocal problem with critical exponent. For and , Wang et al. [27] proved the existence of two positive solutions of with an additional inhomogeneous perturbation on the whole space . When and is replaced by a nonnegative function , [28] showed how the shape of the graph of affects the number of positive solutions to . However, there are no known existence results for provided and .

Motivated by the works described above, in the present paper, we try to prove the existence and multiplicity of positive solutions of problem when for some (see Theorem 1), provide uniform estimates of extremal values for problem (see Theorem 2), and obtain the blow-up and asymptotic behavior of these positive solutions when and (see Theorem 3).

Denote by the standard Sobolev space endowed with the standard norm . Let be the norm of the space . Denote by () the strong (weak) convergence. and denote various positive constants whose exact values are not important. Let be the positive principal eigenvalue of the operator on with corresponding positive principal eigenfunction . Denote by the best constant in the Sobolev embedding , namely,

It is well known that the weak solutions of problem correspond to the critical points of the following energy functional

Moreover, we easily see that .

Define the manifold and decompose into three subsets as follows:

Set

Our main results are as follows.

Theorem 1. Assume that , then problem has at least two positive solutions and with .

Theorem 2. Let Then, we have where is defined as above and

Theorem 3. Assume that and are two sequences satisfying and as . Let and be the two positive solutions of corresponding to and obtained in Theorem 1 with and , then passing to a subsequence if necessary, (i) as (ii) in as , where is a positive ground state solution of the problem

Remark 4. The multiplicity result of with has been proved by [10]. So, our result presented in Theorem 1 can be viewed as an extension of [10] considering the subcritical case where . In particular, we provide uniform estimates of extremal values for the problem, which are observed for the first time in the studies of such nonlocal problem like .

Remark 5. Comparing with [16], which considered problem with , we in this paper investigate the nonlocal case of . Moreover, unlike [22–24, 26], where the nonlocal term is positive, here we study the case of negative sign of nonlocal term and additionally obtain a bound from above for the parameter.

The plan of this paper is as follows. In Section 2, we give some preliminaries. Section 3 is devoted to the Proof of Theorem 1. In Section 4, we prove Theorems 2 and 3. In the proof of our main results, we use variational methods, and they are inspired by [10, 16]. However, in the present paper, we encounter some new difficulties due to the critical growth and nonlocal term. Firstly, compared with [10], the calculations here are more delicate and difficult since we now face the critical problem . Secondly, to provide the bound from above for of involving nonlocal term, we need to develop some techniques applied in [16] where dealt with local case. Thirdly, in order to obtain the asymptotic behavior of the solutions of as in the work of [10], we add the condition of and conduct some new analysis.

2. Preliminaries

Lemma 6. Let . Then, and .

Proof. A simple calculation shows that For any , , set Since , it is clear that and . Moreover, is concave and achieves its maximum at the point with By Hölder and Sobolev inequalities, for , we obtain From which we infer that there exist two constants and satisfying and This gives that and .
In what follows, we prove that . Suppose to the contrary that there is with . By , we have As a consequence, by Sobolev inequality, Moreover, we can also infer from that and so Combining (20) and (21), for , we conclude that which is absurd. The proof of Lemma 6 is completed.

Lemma 7. Assume that , then there is a gap structure in : where

Proof. In the case of , using Sobolev inequality, we have which yields .
In the case of , it holds which gives that . Moreover, we easily check that if , then .

Lemma 8. For any , there exist and a differential functional such that

Proof. Fix and define by Since for , one has and then we can employ the implicit function theorem for at the point and derive and a differential functional defined for , such that In view of the continuity of , we may choose possibly smaller () such that for any , , it holds In a similar way, we can prove the case of , and thus, Lemma 8 follows.

Lemma 9. If , then we have (i)The functional is coercive and bounded from below on (ii)

Proof. (i)For , using Hölder’s inequality, we obtainThis proves the conclusion (i). (ii)For , it holdsCombining this and Lemma 6, we have that . Furthermore, we deduce from (i) that . Thus, .

Lemma 10. If , then and are closed.

Proof. Let be a sequence in such that in . Since , we have namely, . For , it then follows from Lemma 7 that . In turn, we obtain , and so, is closed for . The same argument can prove that is closed. This completes the proof of Lemma 10.

3. Proof of Theorem 1

Lemma 11. Suppose that , then problem admits a positive solution with .

Proof. By Lemmas 9 and 10, we can apply Ekeland variational principle to get a minimizing sequence such that Since , we can assume that in . By Lemma 9, is bounded in , and so, we may assume that In the following, we prove that is a positive solution to . To this purpose, we divide the proof into five steps.
Step 1. .
If, to the contrary, we have . Since , it follows that for large, and hence, which contradicts with (35). Therefore, .
Step 2. There is a positive constant satisfying To prove that, it suffices to check that In view of , one has Assume to the contrary that Then, we can suppose as , where satisfies From this, we have that for , which implies that in , contradicting . In turn, we deduce that (40) holds.
Step 3. as .
Let , , where and are defined as Lemma 8 with . Let with . Fix and set . Since , it follows from (36) that By the definition of Fréchet derivative, we obtain Then, and hence, which yields that From Step 2, Lemma 8, and the boundedness of , we also have As a consequence, for fixed , we can derive letting in (50) that which implies that as .
Step 4. in .
Set . If , we are done, thus assume . By and (37), Moreover, from , the boundedness of , and Brézis-Lieb lemma, we have that Combining this and (53), we get It then follows from Sobolev inequality that Passing the limit as , we obtain that By (53), (57), and Hölder inequality, For and , define By easy calculation, we have that achieves its minimum value at and Therefore, we obtain Using (37), (53), and (61), we deduce that for , which is a contradiction since . This implies that is impossible. Hence, ; that is, in .
Step 5. is a positive solution of problem and .
From (35) and Steps 3 and 4, we have that, up to a subsequence, in with and . Namely, is a weak nontrivial solution of problem . Moreover, by Lemmas 6 and 10, we know . Standard elliptic regularity argument and strong maximum principle provide that is positive. Therefore, the proof of Lemma 11 is completed.

Lemma 12. Let , then problem has a positive solution with .

Proof. As in the proof of Lemma 11, we can prove that there exists a bounded and nonnegative sequence with the properties (i)(ii)(iii)(iv)(v)Without loss of generality, we may assume that . Let be a cut-off function such that in and near zero. Set By [29, 30], one has for small, In the first place, we prove the following upper bound for , where is the positive solution obtained in Lemma 11. Since , it is easy to verify that . By the fact that , we also have and hence, Let with . It follows from (67) that Let be given by Lemma 6. As can be seen from the proof of Lemma 6, there exist and , where . From the structure of and the fact of , we easily see that is uniformly bounded by a suitable constant , , and .
Moreover, we have from (68) that there is satisfying Therefore, we may find such that , , and .
Define Notice that and . Because and the continuity of , we have that for must intersect . As a consequence, Thus, to complete the proof of (65), it suffices to show that By mean value theorem, there exists such that for any . Using (66), (67), and (73), we obtain To proceed, we set Recall that, for , it holds for some . By (73) and (76), we have that which implies that there exists a constant small enough such that Thus, we only need to consider the case of . By the same argument of Lemma 11 of [21], we have Combining this and (64), we have for sufficiently small, where is a positive constant independent of . This together with (74) implies that (65) holds.
In the second place, we claim that . If, to the contrary, we have . Since , it follows that and so, by Sobolev inequality Assume that . By and Lemma 7, we obtain that . Taking in (82), we have , and thus which is a contradiction with (65). Therefore, the claim follows. At this point, we may proceed as in the proof of Lemma 11 and conclude that is a positive solution of problem with . This completes the proof of Lemma 12.

Proof of Theorem 1. Theorem 1 is an immediate consequence of Lemmas 7, 11, and 12.

4. Proofs of Theorems 2 and 3

Proof of Theorem 2. By the definition of and Theorem 1, we easily see that . Hence, Proof of Theorem 2 is completed if we show that . To this goal, let us define the functions Obviously, we have that is convex and attains its minimum at the point with As a consequence, we can take such that This gives that namely, Assume that any is such that admits a positive solution . On the one hand, using (89) with , multiplying by , and integrating over , we get On the other hand, multiplying by and integrating over , there holds Since we infer from (90) and (91) that . By the arbitrariness of and the definition of , we conclude that . Proof of Theorem 2 is thus completed.

Proof of Theorem 3. Let and be two sequences satisfying and as , and let and be the two positive solutions of obtained in Theorem 1 with and .
Using Lemma 7 and , we have that and thus, the conclusion (i) holds.
In what follows, we prove the conclusion (ii) of Theorem 3. Noting that for all , we obtain from Hölder inequality that As a consequence of , we have that is bounded in . Thus, there is a subsequence of (still denoted by ) such that in as . Furthermore, for all , it holds which provides that is a nonnegative weak solution of problem . Let be the corresponding functional of defined by Since as , it follows that in .
Define . It is easy to check that there exists such that and . As for any , we easily see that . Set and suppose that . We claim . Otherwise, we have , and hence, by , as and is bounded in ; one has for large , a contradiction. Thus, the claim follows. Then, which implies that is a global minimum of . This result, together with strong maximum principle proves that is a positive ground state solution of . Theorem 3 is thus proved.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

This research was supported by the Natural Science Foundation of Fujian Province (Nos. 2021J011228 and 2021J01330).

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Copyright © 2022 Zhigao Shi and Xiaotao Qian. 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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