Annales Academię Scientiarum Fennicę
Mathematica
Volumen 38, 2013, 515-534

THE EXISTENCE OF INFINITELY MANY SOLUTIONS FOR p-LAPLACIAN TYPE EQUATIONS ON R^N WITH LINKING GEOMETRY

Gongbao Li and Hongyu Ye

Central China Normal University, School of Mathematics and Statistics
Wuhan 430079, P.R. China; ligb 'at' mail.ccnu.edu.cn

Central China Normal University, School of Mathematics and Statistics
Wuhan 430079, P.R. China; yyeehongyu 'at' yahoo.com.cn

Abstract. In this paper, we study the existence of infinitely many solutions to the following quasilinear equation of p-Laplacian type in R^N

(0.1) -\triangle_p u + |u|^p-2u = \lambda V(x)|u|^p-2u + g(x,u), u \in W^1,p(R^N)

with sign-changing radially symmetric potential V(x), where 1 < p < N, \lambda \in R and \triangle_p u = div(|Du|^p-2Du) is the p-Laplacian operator, g(x,u) \in C(R^N x R,R) is subcritical and p-superlinear at 0 as well as at infinity. We prove that under certain assumptions on the potential V and the nonlinearity g, for any \lambda \in R, the problem (0.1) has infinitely many solutions by using a fountain theorem over cones under Cerami condition. A minimax approach, allowing an estimate of the corresponding critical level, is used. New linking structures, associated to certain variational eigenvalues of -\triangle_p u + |u|^p-2u = \lambda V(x)|u|^p-2u are recognized, even in absence of any direct sum decomposition of W^1,p(R^N) related to the eigenvalue itself.

Our main result can be viewed as an extension to a recent result of Degiovanni and Lancelotti in [10] concerning the existence of nontrivial solutions for the quasilinear elliptic problem:

(0.2) -\triangle_p u = \lambda V(x)|u|^p-2u + g(x,u), in \Omega, u = 0, on \partial\Omega,

where $\Omega\subset\R^N$ is a bounded open domain.

2010 Mathematics Subject Classification: Primary 35J20, 35J60, 35J92.

Key words: Fountain theorem over cones, cohomological index, p-Laplace equation, infinitely many solutions.

Reference to this article: G. Li and H. Ye: The existence of infinitely many solutions for p-Laplacian type equations on R^N with linking geometry. Ann. Acad. Sci. Fenn. Math. 38 (2013), 515-534.

Full document as PDF file

doi:10.5186/aasfm.2013.3830

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