Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities (Q435121)

In this paper, the author considers the existence of solutions to the quasilinear Schrödinger equation \(-\Delta u+V(x) u-\Delta (u^2) u=g(x,u)\) in \(\mathbb R^N\), where \(N\geq 3\) and \( g(x,u)\) is a superlinear but subcritical function. Using Mountain Pass Approach with a weaker form of an Ambrosetti-Rabinowitz condition and under periodic or asymptotically periodic assumptions on the nonlinearities, the author proves the existence of at least one positive solution. Examples illustrating the results are given throughout the paper.