Acta mathematica scientia, Series B >
FOUNTAIN THEOREM OVER CONES AND APPLICATIONS
Received date: 2010-04-22
Online published: 2010-11-20
Supported by
The first author is supported by ARC grant of Australia; the second author is supported by National Natural Sciences Foundations of China (10961016 and 10631030); NSF of Jiangxi (2009GZS0011).
In this paper, we establish fountain theorems over cones and apply it to the quasilinear elliptic problem
{ -Δp u =λ |u|q-2u +μ |u| γ-2u, x∈Ω,
u = 0, x∈∂Ω, （1）
to show that problem (1) possesses infinitely many solutions, where 1<p<N, 1< q <p < γ, Ω（RN is a smooth bounded domain and λ，μ∈R.
YAN Shu-Sen , YANG Jian-Fu . FOUNTAIN THEOREM OVER CONES AND APPLICATIONS[J]. Acta mathematica scientia, Series B, 2010 , 30(6) : 1881 -1888 . DOI: 10.1016/S0252-9602(10)60180-4
[1] Azorero J G, Alonso I Peral. Comportement asymptotique des valeurs propres du p-Laplacien. C R Acad Sci, Paris Sér I Math, 1988, 307(2): 75--78
[2] Bartsch T. Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Analysis, TMA, 1993, 20: 1205--1216
[3] Bartsch T, Willem M. On an elliptic equation with concave and convex nonlinearities. Proc Amer Math Soc, 1995, 123(11): 3555--3561
[4] Cuesta M. Eigenvalue problems for the p-Laplacian with indefinite weights. Electronic J Differential Equations, 2001, 33: 1--9
[5] Cingolani S, Degiovanni M. Nontrivial solutions for p-Laplace equations with right-hand side having p-Linear growth at infinity. Comm Partial Differential Equations, 2005, 30: 1191--1203
[6] Degiovanni M, Lancelotti S. Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity. Ann Inst H
Poincaré Anal Non Linéaire, 2007, 27: 907--919
[7] Degiovanni M, Lancelotti S. Linking solutions for p-Laplace equations with nonlinearity at critical growth. J Funct Anal, 2009, 256(11): 3643--3659
[8] Drábek P, Robinson S B. Resonance problems for the p-Laplacian. J Funct Anal, 1999, 169: 189--200
[9] Fadell E R, Rabinowitz P H. Difurcation for odd potential operators and an alternative topological index. J Funct Anal, 1977, 26(1): 48--67
[10] Fadell E R, Rabinowitz P H. Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for
Hamiltonian systems. Invent Math, 1978, 45(2): 139--174
[11] Frigon M. On a new notion of linking and application to elleptic problem at resonance.J Differential Equations, 1999, 151(1): 96--120
[12] Perera K. Nontrivial critical groups in p-Laplacian problems via the Yang index. Topol Methods Nonlinear Anal, 2003, 21(2): 301--309
[13] Perera K, Szulkin A. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete Contin Dyn Syst, 2005, 13(3): 743--753
[14] Willem M. Minimax Theorems. Boston: Birkhäuser, 1996
[15] Yang Jianfu, Zhu Xiping, On the existence of nontrivial solutions of a quasilinear elliptic boundary value problem for unbounded domains (I). Acta Math Sci, 1987, 7: 341--359
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