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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, 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), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar

[2]

H. Brézis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives, in "Directions in Partial Differential Equations" (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, (1987), 17-36.  Google Scholar

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H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151.  Google Scholar

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. 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), 107-136. doi: 10.1006/jfan.1996.0058.  Google Scholar

[6]

T. Bartsch and D. G. de Figueiredo, Infinitely many solutions for nonlinear elliptic systems, in "Topics in Nonlinear Analysis," Progr. Nonlinear Differential Equations Appl., 35, Birkhäuser, Basel, (1999), 51-67.  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), 90-117. 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), 1857-1866. 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), 2861-2902. 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), 471-501. 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), 1389-1414. 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), 1-18.  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), 2973-2989. 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), 1257-1280.  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), 877-895. 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), 309-334. 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), 1163-1177. doi: 10.1017/S0308210508000875.  Google Scholar

[18]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426. 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), 243-256. doi: 10.1007/s11425-010-4131-3.  Google Scholar

[20]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[21]

M. Willem, "Minimax Theorems," Progr. Nonlinear Differential Equations Appl., 24, Birkhäser Boston, Inc., Boston, MA, 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), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar

[2]

H. Brézis, Nonlinear elliptic equations involving the critical Sobolev exponent-survey and perspectives, in "Directions in Partial Differential Equations" (Madison, WI, 1985), Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, (1987), 17-36.  Google Scholar

[3]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151.  Google Scholar

[4]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. 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), 107-136. doi: 10.1006/jfan.1996.0058.  Google Scholar

[6]

T. Bartsch and D. G. de Figueiredo, Infinitely many solutions for nonlinear elliptic systems, in "Topics in Nonlinear Analysis," Progr. Nonlinear Differential Equations Appl., 35, Birkhäuser, Basel, (1999), 51-67.  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), 90-117. 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), 1857-1866. 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), 2861-2902. 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), 471-501. 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), 1389-1414. 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), 1-18.  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), 2973-2989. 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), 1257-1280.  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), 877-895. 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), 309-334. 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), 1163-1177. doi: 10.1017/S0308210508000875.  Google Scholar

[18]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426. 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), 243-256. doi: 10.1007/s11425-010-4131-3.  Google Scholar

[20]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[21]

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

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