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Infinitely many radial solutions to elliptic systems involving critical exponents
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 |
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. |
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. |
[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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