Localized nodal solutions for quasilinear Schrödinger equations

Abstract

In this paper, we study the existence of localized nodal solutions for a class of semiclassical quasilinear Schrödinger equations including, as a special case, the Modified Nonlinear Schrödinger Equation (MNLS)ε2(Δv+12vΔv2)−V(x)v+|v|q−2v=0,inRN,v(x)→0as|x|→∞. We establish for small ε the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function, by developing new variational perturbation method to treat this class of non-smooth variational problems. The new method allows the perturbed variational functionals to share critical points with the original functional. This method allows us to avoid any limiting process from the perturbed problems to the original problem, and it is effective in dealing with multiple existence of solutions.