A variant of symmetric mountain pass theorem and its applications for generalized quasi-linear elliptic equations

Abstract

By using a variant of the variational perturbation techniques introduced by Rabinowitz [Multiple critical points of perturbed symmetric functionals. Tran Amer Math Soc. 1982;272:753–769], we build a new non-smooth symmetric Mountain Pass Theorem without symmetric condition. This new abstract result can be applied to generalized quasi-linear elliptic equations with small perturbations. Our results extend the results concerning a semi-linear elliptic equation made by Li-Liu [Perturbations from symmetric elliptic boundary value problems. J Differ Equ. 2002;185:271–280].