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Mirror symmetry for a Hessian over-determined problem and its generalization
Infinitely many sign-changing solutions for the Brézis-Nirenberg problem
1. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 |
References:
[1] |
G. Arioli, F. Gazzola, H. C. Grunau and E. Sassone, The second bifurcation branch for radial solutions of the Brézis-Nirenberg problem in dimension four}, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 69-90.
doi: 10.1007/s00030-007-6034-8. |
[2] |
F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations, 85 (1990), 151-170.
doi: 10.1016/0022-0396(90)90093-5. |
[3] |
T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42.
doi: 10.1081/PDE-120028842. |
[4] |
T. Bartsch, Z. L. Liu and T. Weth, Nodal solutions of a $p$-Laplcain equation, Proc. London Math. Soc., 91 (2005), 129-152.
doi: 10.1112/S0024611504015187. |
[5] |
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. |
[6] |
D. M. Cao and S. 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. |
[7] |
D. M. Cao, S. J. Peng and S. 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. |
[8] |
A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéarier, 2 (1985), 463-470. |
[9] |
G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéarier, 1 (1984), 341-350. |
[10] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[11] |
Z. J. Chen and W. M. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987.
doi: 10.1016/j.jde.2011.09.042. |
[12] |
M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem, Adv. Differ. Equ., 10 (2005), 463-480. |
[13] |
G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differ. Equ., 7 (2002), 1257-1280. |
[14] |
G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Comm. Contemp. Math., 5 (2003), 171-177.
doi: 10.1142/S0219199703000938. |
[15] |
D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 205-213.
doi: 10.1017/S0308210500022046. |
[16] |
S. J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227.
doi: 10.1090/S0002-9947-02-03031-3. |
[17] |
P. L. Lions, The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana, 1 (1985), 45-121, 145-201.
doi: 10.4171/RMI/6. |
[18] |
Z. L. Liu, F. A. van Heerden and Z.-Q. Wang, Nodal type bound states of Schrödinger equations via invariant set and minimax methods, J. Differential Equations, 214 (2005), 358-390.
doi: 10.1016/j.jde.2004.08.023. |
[19] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986. |
[20] |
M. Schechter and W. Zou, On the Brézis-Nirenberg problem, Arch. Rational Mech. Anal., 197 (2010), 337-356.
doi: 10.1007/s00205-009-0288-8. |
[21] |
S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space, Ann. Inst. H. Poincaré Anal. Non Linéarier, 12 (1995), 319-337. |
[22] |
M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[23] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000.
doi: 10.1007/978-3-662-04194-9. |
[24] |
A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differential Integral Equations, 22 (2009), 913-926. |
[25] |
S. S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents, Chinese Ann. Math. Ser. A, 16 (1995), 227-234. |
show all references
References:
[1] |
G. Arioli, F. Gazzola, H. C. Grunau and E. Sassone, The second bifurcation branch for radial solutions of the Brézis-Nirenberg problem in dimension four}, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 69-90.
doi: 10.1007/s00030-007-6034-8. |
[2] |
F. V. Atkinson, H. Brézis and L. A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differential Equations, 85 (1990), 151-170.
doi: 10.1016/0022-0396(90)90093-5. |
[3] |
T. Bartsch, Z. L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42.
doi: 10.1081/PDE-120028842. |
[4] |
T. Bartsch, Z. L. Liu and T. Weth, Nodal solutions of a $p$-Laplcain equation, Proc. London Math. Soc., 91 (2005), 129-152.
doi: 10.1112/S0024611504015187. |
[5] |
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. |
[6] |
D. M. Cao and S. 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. |
[7] |
D. M. Cao, S. J. Peng and S. 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. |
[8] |
A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéarier, 2 (1985), 463-470. |
[9] |
G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéarier, 1 (1984), 341-350. |
[10] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[11] |
Z. J. Chen and W. M. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987.
doi: 10.1016/j.jde.2011.09.042. |
[12] |
M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem, Adv. Differ. Equ., 10 (2005), 463-480. |
[13] |
G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differ. Equ., 7 (2002), 1257-1280. |
[14] |
G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Comm. Contemp. Math., 5 (2003), 171-177.
doi: 10.1142/S0219199703000938. |
[15] |
D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A, 105 (1987), 205-213.
doi: 10.1017/S0308210500022046. |
[16] |
S. J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227.
doi: 10.1090/S0002-9947-02-03031-3. |
[17] |
P. L. Lions, The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Iberoamericana, 1 (1985), 45-121, 145-201.
doi: 10.4171/RMI/6. |
[18] |
Z. L. Liu, F. A. van Heerden and Z.-Q. Wang, Nodal type bound states of Schrödinger equations via invariant set and minimax methods, J. Differential Equations, 214 (2005), 358-390.
doi: 10.1016/j.jde.2004.08.023. |
[19] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986. |
[20] |
M. Schechter and W. Zou, On the Brézis-Nirenberg problem, Arch. Rational Mech. Anal., 197 (2010), 337-356.
doi: 10.1007/s00205-009-0288-8. |
[21] |
S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space, Ann. Inst. H. Poincaré Anal. Non Linéarier, 12 (1995), 319-337. |
[22] |
M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[23] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, $3^{rd}$ Edition, Springer-Verlag, Berlin, 2000.
doi: 10.1007/978-3-662-04194-9. |
[24] |
A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differential Integral Equations, 22 (2009), 913-926. |
[25] |
S. S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents, Chinese Ann. Math. Ser. A, 16 (1995), 227-234. |
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