American Institute of Mathematical Sciences

Advanced
February  2014, 34(2): 461-475. doi: 10.3934/dcds.2014.34.461

Infinitely many radial solutions to elliptic systems involving critical exponents

Yinbin Deng 1, , Shuangjie Peng 2, and  Li Wang 3,

1. 

Department of Mathematics, Huazhong Normal University, Wuhan 430079
2. 

Department of Mathematics, Central China Normal University, Wuhan 430079
3. 

School of Basic Science, East China Jiaotong University, Nanchang 330013, China

Received  May 2012 Revised  June 2013 Published  August 2013

In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
Keywords: elliptic systems, Radial solution, (PS) condition., critical exponent.
Mathematics Subject Classification: 35B33, 35J6.
Citation: Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461
References:
[1]

A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519. doi: 10.1006/jfan.1994.1078. Google Scholar

[2]

H. Brézis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives,, in, 54 (1987), 17. Google Scholar

[3]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials,, J. Math. Pures Appl. (9), 58 (1979), 137. Google Scholar

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

[5]

T. Bartsch and M. Clapp, Critical point theory for indefinite functionals with symmetries,, J. Funct. Anal., 138 (1996), 107. doi: 10.1006/jfan.1996.0058. Google Scholar

[6]

T. Bartsch and D. G. de Figueiredo, Infinitely many solutions for nonlinear elliptic systems,, in, 35 (1999), 51. Google Scholar

[7]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u =u^{\frac{N+2}{N-2}}$ in $\mathbbR^N$,, J. Funct. Anal., 88 (1990), 90. doi: 10.1016/0022-1236(90)90120-A. Google Scholar

[8]

D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent,, Proc. Amer. Math. Soc., 131 (2003), 1857. doi: 10.1090/S0002-9939-02-06729-1. Google Scholar

[9]

D. Cao, S. Peng and S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth,, J. Funct. Anal., 262 (2012), 2861. doi: 10.1016/j.jfa.2012.01.006. Google Scholar

[10]

D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential,, Calc. Var. Partial Differential Equations, 38 (2010), 471. doi: 10.1007/s00526-009-0295-5. Google Scholar

[11]

D. Cao and S. Yan, Infinitely many solutions for an elliptic Neumann problem involving critical Sobolev growth,, J. Differential Equations, 251 (2011), 1389. doi: 10.1016/j.jde.2011.05.011. Google Scholar

[12]

M. Clapp, Y. Ding and S. Hernández-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems,, Electron. J. Differential Equations, 100 (2004), 1. Google Scholar

[13]

D. G. de Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems,, Trans. Amer. Math. Soc., 355 (2003), 2973. doi: 10.1090/S0002-9947-03-03257-4. Google Scholar

[14]

G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth,, Adv. Differential Equations, 7 (2002), 1257. Google Scholar

[15]

J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term,, Trans. Amer. Math. Soc., 323 (1991), 877. doi: 10.2307/2001562. Google Scholar

[16]

M. Grossi, A class of solutions for the Neumann problem $-\Delta u + \lambda u = u^{\frac{N+2}{N-2}}$,, Duke Math. J., 79 (1995), 309. doi: 10.1215/S0012-7094-95-07908-3. Google Scholar

[17]

T. Hsu and H. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1163. doi: 10.1017/S0308210508000875. Google Scholar

[18]

E. Jannelli, The role played by space dimension in elliptic critical problems,, J. Differential Equations, 156 (1999), 407. doi: 10.1006/jdeq.1998.3589. Google Scholar

[19]

D. Kang and S. Peng, Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents,, Sci. China Math., 54 (2011), 243. doi: 10.1007/s11425-010-4131-3. Google Scholar

[20]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. doi: 10.1007/BF02418013. Google Scholar

[21]

M. Willem, "Minimax Theorems,", Progr. Nonlinear Differential Equations Appl., 24 (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

show all references

References:
[1]

A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519. doi: 10.1006/jfan.1994.1078. Google Scholar

[2]

H. Brézis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives,, in, 54 (1987), 17. Google Scholar

[3]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials,, J. Math. Pures Appl. (9), 58 (1979), 137. Google Scholar

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

[5]

T. Bartsch and M. Clapp, Critical point theory for indefinite functionals with symmetries,, J. Funct. Anal., 138 (1996), 107. doi: 10.1006/jfan.1996.0058. Google Scholar

[6]

T. Bartsch and D. G. de Figueiredo, Infinitely many solutions for nonlinear elliptic systems,, in, 35 (1999), 51. Google Scholar

[7]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u =u^{\frac{N+2}{N-2}}$ in $\mathbbR^N$,, J. Funct. Anal., 88 (1990), 90. doi: 10.1016/0022-1236(90)90120-A. Google Scholar

[8]

D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent,, Proc. Amer. Math. Soc., 131 (2003), 1857. doi: 10.1090/S0002-9939-02-06729-1. Google Scholar

[9]

D. Cao, S. Peng and S. Yan, Infinitely many solutions for $p$-Laplacian equation involving critical Sobolev growth,, J. Funct. Anal., 262 (2012), 2861. doi: 10.1016/j.jfa.2012.01.006. Google Scholar

[10]

D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential,, Calc. Var. Partial Differential Equations, 38 (2010), 471. doi: 10.1007/s00526-009-0295-5. Google Scholar

[11]

D. Cao and S. Yan, Infinitely many solutions for an elliptic Neumann problem involving critical Sobolev growth,, J. Differential Equations, 251 (2011), 1389. doi: 10.1016/j.jde.2011.05.011. Google Scholar

[12]

M. Clapp, Y. Ding and S. Hernández-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems,, Electron. J. Differential Equations, 100 (2004), 1. Google Scholar

[13]

D. G. de Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems,, Trans. Amer. Math. Soc., 355 (2003), 2973. doi: 10.1090/S0002-9947-03-03257-4. Google Scholar

[14]

G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth,, Adv. Differential Equations, 7 (2002), 1257. Google Scholar

[15]

J. Garcia Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term,, Trans. Amer. Math. Soc., 323 (1991), 877. doi: 10.2307/2001562. Google Scholar

[16]

M. Grossi, A class of solutions for the Neumann problem $-\Delta u + \lambda u = u^{\frac{N+2}{N-2}}$,, Duke Math. J., 79 (1995), 309. doi: 10.1215/S0012-7094-95-07908-3. Google Scholar

[17]

T. Hsu and H. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1163. doi: 10.1017/S0308210508000875. Google Scholar

[18]

E. Jannelli, The role played by space dimension in elliptic critical problems,, J. Differential Equations, 156 (1999), 407. doi: 10.1006/jdeq.1998.3589. Google Scholar

[19]

D. Kang and S. Peng, Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents,, Sci. China Math., 54 (2011), 243. doi: 10.1007/s11425-010-4131-3. Google Scholar

[20]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353. doi: 10.1007/BF02418013. Google Scholar

[21]

M. Willem, "Minimax Theorems,", Progr. Nonlinear Differential Equations Appl., 24 (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[1]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[2]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[3]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[4]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[5]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[6]

Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698
[7]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117
[8]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[9]

João Marcos do Ó, Sebastián Lorca, Justino Sánchez, Pedro Ubilla. Positive radial solutions for some quasilinear elliptic systems in exterior domains. Communications on Pure & Applied Analysis, 2006, 5 (3) : 571-581. doi: 10.3934/cpaa.2006.5.571
[10]

Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531
[11]

Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013
[12]

Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715
[13]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[14]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[15]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[16]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697
[17]

Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
[18]

Haiyang He, Jianfu Yang. Positive solutions for critical elliptic systems in non-contractible domains. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1109-1122. doi: 10.3934/cpaa.2008.7.1109
[19]

Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859
[20]

Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences