Acta mathematica scientia, Series B >
SOLUTIONS FOR THE QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY--SOBOLEV EXPONENTS
Received date: 2008-07-16
Online published: 2010-09-20
Supported by
This work is supported partly by the National Natural Science Foundation of China (10771219).
In this article, we study the quasilinear elliptic problem involving critical Hardy--Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign--changing solutions to the problem.
Key words： quasilinear problem; critical exponent; solution; variational method
KANG Dong-Sheng . SOLUTIONS FOR THE QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY--SOBOLEV EXPONENTS[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1529 -1540 . DOI: 10.1016/S0252-9602(10)60146-4
[1] Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequality with weights. Compos Math, 1984, 53: 259--275
[2] Brezis H, Nirenberg L. {Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents}. Comm Pure Appl Math, 1983, 36: 437--477
[3] Kang D. On the quasilinear elliptic problems with critical Sobolev--Hardy exponents and Hardy terms. Nonlinear Anal, 2008, 68: 1973--1985
[4] Cao D, Han P. Solutions to critical elliptic equations with multi--singular inverse square potentials. J Differ Equ, 2006, 224: 332--372
[5] Cao D, Peng S. A note on the sign--changing solutions to elliptic problems with critical Sobolev and Hardy terms. J Differ Equ, 2003, 193: 424--434
[6] CatrinaF, Wang Z Q. {On the Caffarelli--Kohn--Nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extermal functions}. Comm Pure Appl Math, 2001, 54: 229--258
[7] Deng Y, Gao Q, Zhang D. {Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on RN. Discrete Contin Dyn Syst, 2006, 14: 707--719
[8] Deng Y, Jin L. Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents. Nonlinear Anal, 2007, 67: 3261--3275
[9] Deng Y, Jin L. On symmetric solutions of a singular elliptic equation with critical Sobolev--Hardy exponent. J Math Anal Appl, 2007, 329: 603--616
[10] Felli V, Terracini S. Nonlinear Schr\"{o}dinger equations with symmetric multi-polar potentials. Calc Var Partial Differ Equ, 2006, 27: 25--58
[11] Ferrero A, Gazzola F. Existence of solutions for singular critical growth semilinear elliptic equations. J Differ Equ, 2001, 177: 494--522
[12] Jannelli E. The role played by space dimension in elliptic critcal problems. J Differ Equ, 1999, 156: 407--426
[13] Smets D. Nonlinear Schr\"{o}dinger equations with Hardy potential and critical nonlinearities. Trans Amer Math Soc, 2005, 357: 2909--2938
[14] Terracini S. On positive solutions to a class of equations with a singular coefficient and critical exponent. Adv Differ Equ, 1996, 1: 241--264
[15] Ghoussoub N, Yuan C, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352: 5703--5743
[16] Han P. Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal, 2005, 61: 735--758
[17] Kang D. Some properties of solutions to the singular quasilinear problems. Nonlinear Anal, 2010, 72: 682--688
[18] Chen Z, Shen Y. Hardy--Sobolev inequalities with general weiths and remaider terms. Acta Mathematica Scientia, 2008, 28B(3): 469--478
[19] Yao Y, Shen Y. On critical singular quasilinear elliptic problem when n=p. Acta Mathematica Scientia, 2006, 26B(2): 209--219
[20] Tarantello G. Nodal solutions of semilinear elliptic equations with critical exponent. Differential Integral Equations, 1992, 5: 25--42
[21] Lions P L. The concentration compactness principle in the calculus of variations, the limit case (I). Rev Mat Iberoamericana, 1985, 1(1): 145--201
[22] Lions P L. The concentration compactness principle in the calculus of variations, the limit case (II). Rev Mat Iberoamericana, 1985, 1(2): 45--121
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