Existence of solution for quasilinear Schrödinger equations with general nonlinear terms and non-compact potentials

Abstract

In this paper, we investigate the existence of solution for a class of quasilinear Schrödinger equations with sub-cube growth nonlinear terms g and the non-compact potential V: $-\Delta u+V\left(x\right)u-u\Delta \left(u^2\right)=g\left(u\right),x\in R^3$. We primarily overcome the difficulties caused by using a perturbation approach together with a new version of global compactness lemma. We establish a globally decomposed version of solution sequences, which is almost novel for this type of problem.