The existence of infinitely many solutions for p-Laplacian type equations on R^N with linking geometry

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) −△pu + |u| p 2 u = V (x)|u| p 2 u + g(x;u); u ∈ W 1;p (R N ) with sign-changing radially symmetric potential V (x), where 1 < p < N; ∈ R and △pu = div(|Du| p 2 Du) is the p-Laplacian operator, g(x;u) ∈ C(R N ×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 ∈ 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 −△pu + |u| p 2 u = V (x)|u| p 2 u 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: