Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities

In this paper, we consider the existence of solutions for the quasilinear Schrödinger equation: −Δu+V(x)u−Δ(u2)u=g(x,u) in RNRN, where N≥3N≥3 and g(x,u)g(x,u) is a superlinear but subcritical function. Under periodic or asymptotically periodic assumptions on the nonlinearities, we prove the existence of bounded states for this equation by the Mountain Pass Theorem.