Acta mathematica scientia, Series B >
INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRÉZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL
Received date: 2014-10-30
Revised date: 2015-10-22
Online published: 2016-04-25
Supported by
Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.
In this article, we give a new proof on the existence of infinitely many sign-changing solutions for the following Brézis-Nirenberg problem with critical exponent and a Hardy potential -Δu-μu/|x|2=λu+|u|2*-2u in Ω, u=0 on ∂Ω, where Ω is a smooth open bounded domain of RN which contains the origin, 2*=2N/N-2 is the critical Sobolev exponent. More precisely, under the assumptions that N≥7, μ∈[0, μ-4), and μ=(N-2)2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ>0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.
Jing ZHANG , Shiwang MA . INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRÉZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL[J]. Acta mathematica scientia, Series B, 2016 , 36(2) : 527 -536 . DOI: 10.1016/S0252-9602(16)30018-2
[1] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14:349-381
[2] Atkinson F V, Brezis H, Peletier L A. Nodal solutions of elliptic equations with critical Sobolev exponents. J Differential Equations, 1997, 134:1-25
[3] Bartsch T, Liu Z, Weth T. Sign changing solutions of superlinear Schrödinger equations. Commun Partial Differential Equations, 2004, 29:25-42
[4] Bartsch T, Liu Z, Weth T. Nodal solutions of a p-Laplcain equation. Proc London Math Soc, 2005, 91:129-152
[5] Cao D, Han P. Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J Differential Equations, 2004, 205:521-537
[6] Cao D, Peng S. A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms. J Differential Equations, 2003, 193:424-434
[7] Cao D, Yan S. Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential. Calc Var Partial Differnential Equations, 2010, 38:471-501
[8] Cao D, Peng S, Yan S. Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth. J Funct Anal, 2012, 262:2861-2902
[9] Chen Z, ZouW. On an elliptic problem with critical exponent and Hardy potential. J Differential Equations, 2012, 252:969-987
[10] Devillanova G, Solomini S. Concentration estimates and multiple solutions to elliptic problems at critical growth. Adv Differ Equ, 2002, 7:1257-1280
[11] Egnell E. Elliptic boundary value problems with singular coefficients and critical nonlinearities. Indiana Univ Math J, 1989, 38:235-251
[12] Ferrero A, Gazzola F. Existence of solutions for singular critical growth semilinear elliptic equations. J Differential Equations, 2001, 177:494-522
[13] Li S, Wang Z Q. Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Trans Amer Math Soc, 2002, 354:3207-3227
[14] Liu Z, Heerden F van, Wang Z Q. Nodal type bound states of Schrödinger equations via invariant set and minimax methods. J Differential Equations, 2005, 214:358-390
[15] Jannelli E. The role played by space dimension in elliptic critical problem. J Differential Equations, 1999, 156:407-426
[16] Rabinowitz P. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence, RI:CBMS Series, 1986, 65
[17] Schechter M, Zou W. On the Brézis-Nirenberg problem. Arch Rational Mech Anal, 2010, 197:337-356
[18] Struwe M. Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. 3rd Edition. Berlin:Springer-Verlag, 2000
[19] Sun J, Ma S. Infinitely many sign-changing solutions for the Brézis-Nirenberg problem. Commun Pure Appl Anal, 2014, 13:2317-2330
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