Infinitely many radial solutions to elliptic systems involving critical exponents

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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

Abstract
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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 and 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.
[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.
[3]	H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.
[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.
[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.
[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.
[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.
[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.
[20]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.
[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.

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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.
[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.
[3]	H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.
[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.
[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.
[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.
[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.
[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.
[20]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.
[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.

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