Radially Symmetric Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators in an Orlicz-Sobolev Space Setting

Abstract

We investigate the following elliptic equations: $$\left\{ {\matrix{ { - M\left( {\int_{{\mathbb{R}^N}} {\phi ({{\left| {\nabla u} \right|}^2}){\rm{d}}x} } \right){\rm{div(}}\phi \prime ({{\left| {\nabla u} \right|}^2})\nabla u{\rm{) + }}{{\left| u \right|}^{\alpha - 2}}u = \lambda h(x,u),} \hfill \cr {u(x) \to 0,\;\;\;\;\;{\rm{as}}\left| x \right| \to \infty ,} \hfill \cr } } \right.\;\;\;\;\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{R}^N},$$ where N ≥ 2, 1 < p < q < N, α < q, 1 < α < p*q†/p† with $${p^ * } = {\textstyle{{Np} \over {N - p}}},\;\;\phi (t)$$ behaves like tq/2 for small t and tp/2 for large t, and p′ and q′ are the conjugate exponents of p and q, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.