Multiple Solutions of Quasilinear Elliptic Equations in ℝN

Abstract

Assume that $Q$ is a positive continuous function in ${\mathbb{ℝ}}^N$ and satisfies some suitable conditions. We prove that the quasilinear elliptic equation $-{\Delta }_{p}u+|u{|}^{p-2}u=Q(z\left)\right|u{|}^{q-2}u$ in ${\mathbb{ℝ}}^N$ admits at least two solutions in ${\mathbb{ℝ}}^N$ (one is a positive ground-state solution and the other is a sign-changing solution).