Multiple Nontrivial Solutions for a Class of Biharmonic Elliptic Equations with Sobolev Critical Exponent

Qian, Xiaoyong; Wang, Jun; Zhu, Maochun

doi:https://doi.org/10.1155/2018/8212785

Mathematical Problems in Engineering

On this page

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

Research Article | Open Access

Volume 2018 | Article ID 8212785 | https://doi.org/10.1155/2018/8212785

Multiple Nontrivial Solutions for a Class of Biharmonic Elliptic Equations with Sobolev Critical Exponent

Xiaoyong Qian,¹Jun Wang,¹and Maochun Zhu¹

Academic Editor: Mariano Torrisi

Received11 Sept 2018

Accepted01 Nov 2018

Published21 Nov 2018

Abstract

In this paper, we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic equation with Sobolev critical exponent in a bounded domain. By using the idea of the previous paper, we generalize the results and prove the existence and multiplicity of nontrivial solutions of the biharmonic elliptic equations.

1. Introduction and Main Results

In the present paper, we are concerned with the existence of multiple solutions to the following biharmonic elliptic equation with perturbationwhere is a bounded domain in , is the biharmonic operator, and is the Sobolev critical exponent.

The second-order semilinear and quasilinear problems have been object of intensive research in the last years. Brezis and Nirenberg [1] have studied the existence of positive solutions of (1). Particularly, when , where is a constant, they have discovered the following remarkable phenomenon: the qualitative behavior of the set of solutions of (1) is highly sensitive to , the dimension of the space. Precisely, Brezis and Nirenberg [1] have shown that, in dimension , there exists a positive solution of (1), if and only if ; while, in dimension and when is the unit ball, there exists a positive solution of (1), if and only if , where is the first eigenvalue of in . For more results on this direction we refer the readers to [2–5] and the references therein.

During the last decades many works have been orientated to the analysis of biharmonic nonlinear Schrödinger equation (BHNSE) is an open domain . For instance, paper [6] proved that some of the properties and characteristics for the second-order semilinear problems can be extended to BHLSE. Paper [7] proved the existence of blow-up solutions. In papers [8–10], the authors proved the existence of global solutions, in particular, looking for standing wave solutions for (2) of the formsuch that is a solution satisfying the equationIf and , we know that (4) admits no positive solutions if is star shaped under the Navier or Dirichlet boundary conditions (see [11, Theorem 3.3] and [12, Corollary 1]). If and is a ball, paper [13] proved the existence of positive radially symmetric solutions. For more general results on this direction one can refer to [14, 15, 15–21] and the references therein.

Motivated by the above results, we study the case that , , and is a bounded domain. Precisely, we shall generalize the results of Tarantello [22] to the biharmonic and critical exponent case. Our main tool here is the Nehari manifold method which is similar to the fibering method of Pohozaev’s.

In order to state the main results, we shall give some notation and assumptions. Let , and be the usual norm. Obviously, is a Hilbert space under the inner product . Correspondingly, the norm is denoted by ; i.e., . Assume that satisfieswhere is the best Sobolev embedding constant of , andLetbe an extremal function for the Sobolev inequality in . For , let and with and near . We point out that the embedding is not compact. This leads to the lack of compactness for the proved existence and multiplicity of nontrivial solutions of (1). Motivated by [1, 22], we recover the local compactness by dividing the Nehari manifold into three parts and give some estimates for the least energy of (1)

It is easy to see that the energy functional of (1) is denoted byHence, is well defined (under (5)) and of the class . Moreover, all the critical points of are precisely the solutions of (1). We define the Nehari manifold associated with the functional by It is clear that all critical points lie in the Nehari manifold, and it is usually effective to consider the existence of critical points in this smaller subset of the Sobolev space. For fixed , we setThe mapping is called fibering map. Such maps are often used to investigate Nehari manifolds for various semilinear problems. From the relationship between and , we can divide into three parts It turns out that under the assumption (5), we infer that (see Lemma 5 below). Now the main result in this paper can be stated as follows.

Theorem 1. Assume that satisfies (5). Thenis achieved at a point . Furthermore, is a critical point of , and when .

In the following we study the second infimum problemIn this case we have the following results.

Theorem 2. Assume that satisfies (5). Then and the infimum in (13) is achieved at a point , which is a critical point of .

The proofs of Theorems 1–2 rely on the Ekeland’s variational principle and careful estimates (see [1]) of minimizing sequence.

2. Some Preliminary Results

In this section we prove some preliminary results for the proof of Theorems 1–2. The main ideas are coming from [1, 22]. We begin with the following lemma which states the purpose of assumption (5).

Lemma 3. Supposed that satisfies (5). For every , there exists a unique such that . Particularly, we haveand . Moreover, if , then there exists a unique such that . In particular, one has and .

Proof. Recall that the fibering map is defined by Then We deduce from that If , we have , and if , one sees . A direct computation shows that achieves its maximum at , and We divide the following two cases to accomplish our results.
Case 1. If , then . It is easy to see that if , we have . So, there exists unique such that and . We infer from the monotonicity of that, for , This shows that .
Case 2. If , we infer from assumption (5) that . Then . Since , there exists a unique such that and . A direct computation shows that and .

Lemma 4. Assume that satisfies (5). We infer that the infimumis achieved, where .

The proof of Lemma 4 is technical and the idea of the proof is mainly motivated by paper [23]. We shall prove it in the appendix. Next we study the property of the set .

Lemma 5. Let satisfy (5). Then for every , we can get the conclusion that .

Proof. We use the contradiction arguments. Assume that, for some , we have . That is,Since , it follows that . Hence, we getBy Sobolev inequality, we deduce that . For , we setFor and , a direct computation shows that We derive from Lemma 4 that, for ,Let . We infer from (26) that which is a contradiction.

Lemma 6. Let satisfy (5). For each , there exist and a differentiable function , satisfying the following:

Proof. We define by Since and (Lemma 5), by using the implicit function theorem at the point we know that the results of the lemma hold.

3. Proof of Theorem 1

In this part we shall give the proof of Theorem 1.

Proof of Theorem 1. We first claim that the functional is bounded from below in . For , we have . That is, . One deduces from (2) and Hölder inequality thatHence, we know that the infimum is also bounded from below. Second, we can get an upper bound for . Let be the solution for . For , one obtains that Set as defined by Lemma 3. Thus, we have that andFor any minimizing sequence , we can use Ekeland’s variational principle (see [24]) to get following properties:(i),(ii). Hence for large enough, we obtain This implies Since , we infer from Hölder’s inequality thatAt the same time, we observe thatOne deduces from (5) and Hölder’s and Sobolev’s inequalities thatSo we derive from (35) and (37) thatwhere and only depend on and .
Next we shall prove that , as . Applying Lemma 6 with and , we can find some such that By condition we haveDividing by and letting , we getwhere . So, we conclude thatwhere is a constant. In order to complete the proof we need to prove that is bounded uniformly on . By Lemma 6 we can getThus, there exists subsequence (still denote by ) such thatWe infer from thatBy the estimate of from (38), we have that andThis is impossible. So, is away from zero. Thus, we conclude thatLet be the weak limit in of . From (47) we can get that is a weak solution for (1). In fact, and So, we have that strongly in and . Moreover, . By using standard method, we can prove that is a global minimum for in (See [25]).

4. Proof of Theorem 2

In this section, we shall give the proof of Theorem 2. Since the embedding is not compact, we need to find some way to recover this compactness. Motivated by previous works of [1, 22, 23], we will seek the level in which -condition will recover. Then we shall use the Mountain-Pass principle to get the second nontrivial solution of (1). The related problems have been studied in [23], and such an approach has been used. The threshold is found in the following lemma to obtain the compactness.

Lemma 7. Assume that the sequence satisfying(i) with , where is defined in (12).(ii) as . Then has a convergent subsequence.

Proof. It is clear that is uniformly bounded from condition and . For a subsequence of , we can get a such thatSo, from , we obtain thatThen is a weak solution of (1), , and , . Let . So, in . By Brezis-Lieb lemma (see [24]), we conclude that Thus, for large enough, we get which meansMoreover, we infer from condition thatand then we obtainNext we shall prove that if (53) and (55) hold, then there exists the subsequence of (still denoted by ), which satisfies We assume is bounded away from 0; that is So from (55) we can get We infer from (53) and (55) that for large. This is contradiction. So, we can get strongly in .

Note that . Following [23], we set to be a set of positive measures such that on . Let us definewhere and are defined in Section 1. Without loss of generality, we take . Then we have the following estimates for .

Lemma 8. and , there exists such thatfor every

Proof. Let and . By the definition of , we can get the Sobolev embedding exponent . A direct computation shows that Now we take the function such thatwhere . On the other hand, we see thatSo, by direct computation we infer thatwhere is the measure of the unit sphere in . Moreover, we have that Thus, we infer from [23] that From all of the above, noticing that , one has thatBy using an estimate obtained by G. Folland [26] and setting outside , one gets that whereConsequently, we have We setand assume achieves its maximum at , which satisfiesWe definewhich is the maximum point of . We can conclude that , and . Let . It is easy to see that . From (73) we can get and then expanding for , we can getSo, one sees that When we take small , we arrive at This finishes the proof.

Now we are ready to give the proof of Theorem 2.

Proof of Theorem 2. It is clear that the uniqueness of satisfies the following condition: At the same time, is a continuous function of . And divides into two components and , which are disconnected from each other. Let Obviously, , and we can check , . We can choose a constant , which satisfiesand claim thatwhere In fact, a direct computation shows that for small enough. Thus, claim (82) holds.
We fix such that both (61) and (82) hold by the choice of and . We setand take , which belongs to . From Lemma 7, we conclude that Since every intersects , we get thatNext we use Mountain-Pass lemma to prove Theorem 2. Let be such that We deduce from Lemma 7 that there exists a subsequence (still denoted by ) of , and such that So, is a critical point for , and .

Remark 9. We point out that the results of Theorems 1–2 can be generalized to polyharmonic problem. Precisely, we can consider the semilinear polyharmonic problemwhere is a smooth bounded domain in . denotes the critical Sobolev exponent for , and is small enough. We can define the energy functional: where is Hilbert space and endowed with the scalar productand is the corresponding norm. Let be an extremal function for the Sobolev inequality in , and the constant be independent of . By dividing the Nehari manifold, we can prove condition when , where and is the first solution. By using the same idea of this article, one can obtain that (89) has at least two nontrivial solutions.

Appendix

In this appendix we mainly focus on the proof of Lemma 4.

Proof of Lemma 4. For , we define Let be the minimizing sequence of (21) with . That is, we have that and in , a.e in and . If , then the conclusion holds. In the following we consider the case by using contradiction argument. Let . So, in . From Brezis-Lieb lemma [27], we obtain that By Sobolev’s inequality, we conclude that Hence we getFrom paper [23], we know that for every , and , there exists such thatwhere is defined in (60). We infer from (A.6) that Thus, for each and , we obtain thatCombining (A.5) and (A.9), we getMoreover, for each one hasThat is,Let and . Then (A.12) implies that is the weak solution ofSince , we can conclude that . Recall that , , and . Replace with , and with if necessarily. For , we take such that We obtain the contradiction if we prove that for a suitable choice of and small .
From (A.7), we infer that as , where . Let , where as . A direct computation shows thatwhere . We deduce from (A.10) and (A.16) and the definition of thatand, furthermore, we infer from (A.16) that Also, we notice that Hence it follows thatThis finishes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Xiaoyong Qian was devoted to prove the first solution of the equation. Jun Wang proved the existence of the second solution of the equation. Maochun Zhu participated in the proof of the section solution of the equation. All authors read and approved the final manuscript.

Acknowledgments

X.-Y. Qian was supported by Jiangsu Province ordinary university graduate student scientific research innovation projects (KYLX 16_0898). J. Wang was supported by NSF of China (Grants 11571140, 11371090), NSF for Outstanding Young Scholars of Jiangsu Province (BK20160063), and NSF of Jiangsu Province (BK20150478) and the Six big talent peaks project in Jiangsu Province (XYDXX-015). M.-C. Zhu was supported by NSF of China (11601190), NSF of Jiangsu Province (BK20160483), and Jiangsu University Foundation Grant (16JDG043).

References

H. Brezis and L. Nirenberg, “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,” Communications on Pure and Applied Mathematics, vol. 36, no. 4, pp. 437–477, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
S. Wang, “The existence of a positive solution of semilinear elliptic equations with limiting Sobolev exponent,” Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 117, no. 1-2, pp. 75–88, 1991.
View at: Publisher Site | Google Scholar | MathSciNet
V. Benci and G. Cerami, “Existence of positive solutions of the Equation in ,” Journal of Functional Analysis, vol. 88, no. 1, pp. 90–117, 1990.
View at: Publisher Site | Google Scholar | MathSciNet
M. Cuesta and C. De Coster, “Superlinear critical resonant problems with small forcing term,” Calculus of Variations and Partial Differential Equations, vol. 54, no. 1, pp. 349–363, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
F. Faraci and C. Farkas, “A quasilinear elliptic problem involving critical Sobolev exponents,” Collectanea Mathematica, vol. 66, no. 2, pp. 243–259, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
G. Fibich, B. Ilan, and G. Papanicolaou, “Self-focusing with fourth-order dispersion,” SIAM Journal on Applied Mathematics, vol. 62, no. 4, pp. 1437–1462, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
G. Baruch, G. Fibich, and E. Mandelbaum, “Singular solutions of the biharmonic nonlinear Schrödinger equation,” SIAM Journal on Applied Mathematics, vol. 70, no. 8, pp. 3319–3341, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
C. Miao, G. Xu, and L. Zhao, “Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case,” Journal of Differential Equations, vol. 246, no. 9, pp. 3715–3749, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
B. Pausader, “Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case,” Dynamics of Partial Differential Equations, vol. 4, no. 3, pp. 197–225, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
B. Pausader, “The cubic fourth-order Schrodinger equation,” Journal of Functional Analysis, vol. 256, no. 8, pp. 2473–2517, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
E. Mitidieri, “A Rellich type identity and applications,” Communications in Partial Differential Equations, vol. 18, no. 1-2, pp. 125–151, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
P. Oswald, “On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball,” Commentationes Mathematicae, vol. 26, no. 3, pp. 565–577, 1985.
View at: Google Scholar | MathSciNet
F. Gazzola, H. Grunau, and M. Squassina, “Existence and nonexistence results for critical growth biharmonic elliptic equations,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 2, pp. 117–143, 2003.
View at: Publisher Site | Google Scholar
C. O. Alves and J. M. do Ó, “Positive solutions of a fourth-order semilinear problem involving critical growth,” Advanced Nonlinear Studies, vol. 2, no. 4, pp. 437–458, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
T. Bartsch, T. Weth, and M. Willem, “A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 3, pp. 253–268, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-X. Ge, “Positive solutions in semilinear critical problems for polyharmonic operators,” Journal de Mathématiques Pures et Appliquées, vol. 84, no. 2, pp. 199–245, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Ge, J. Wei, and F. Zhou, “A critical elliptic problem for polyharmonic operators,” Journal of Functional Analysis, vol. 260, no. 8, pp. 2247–2282, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Deng and W. Shuai, “Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity,” Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 145, no. 2, pp. 281–299, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
T. Bartsch, M. Schneider, and T. Weth, “Multiple solutions of a critical polyharmonic equation,” Journal für die Reine und Angewandte Mathematik. [Crelle's Journal], vol. 571, pp. 131–143, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
F. Bernis, J. García Azorero, and I. Peral, “Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order,” Advances in Differential Equations, vol. 1, no. 2, pp. 219–240, 1996.
View at: Google Scholar | MathSciNet
H.-C. Grunau, “Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,” Calculus of Variations and Partial Differential Equations, vol. 3, no. 2, pp. 243–252, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
G. Tarantello, “On nonhomogeneous elliptic equations involving critical Sobolev exponent,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, vol. 9, no. 3, pp. 281–304, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
H. Brezis and L. Nirenberg, “A minimization problem with critical exponent and nonzero data,” in Symmetry in Nature, Scuola Norm. Sup, vol. 1, pp. 129–140, Scuola Norm. Sup, Pisa, 1989.
View at: Google Scholar
M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996.
View at: Publisher Site | MathSciNet
M. Badiale and E. Serra, Semilinear elliptic equations for beginners, Universitext, Springer, London, 2011.
View at: Publisher Site | MathSciNet
G. B. Folland, Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, New York, NY, USA, 1984.
View at: MathSciNet
H. Brézis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2018 Xiaoyong Qian et al. 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.

PDF Download Citation

Download other formats

Order printed copies