Multiple solutions for perturbed quasilinear elliptic problems with oscillatory terms (Q690948)

The following \(P_{\epsilon}\) problems are considered: \[ -div(|\nabla u|^{p-2}\nabla u)+|u|^{p-2}u=Q(x)(f(u)+\epsilon g(u)); \, x \in R^N; \] \(u(x)\rightarrow 0\) for large \(|x|\); \(p\) is a positive real number; \(Q\) maps \(R^N\) to \(R\); \(Q \in L^1(R^N) \cap L_{\infty}(R^N)\). The results of \textit{A. Kristály} [J. Differ. Equations 245, No. 12, 3849--3868 (2008; Zbl 1169.35025)] are improved, by removing the assumption \(Q\) is radial (no symmetry properties are imposed for \(Q\)). The Palais-Smale condition and the Mountain-Pass theorem are verified for a corresponding functional, to prove the existence of multiple non-negative solutions of \(P_{\epsilon}\), when \(f\) and \(g\) are continuous, \(f(0)=g(0)=0\) and \(f\) oscillates at the origin or at infinity.