On a class of quasilinear Schrödinger equations with vanishing potentials and mixed nonlinearities

Abstract

In this paper, we study the following generalized quasilinear Schrodinger equations with mixed nonlinearity $$\left\{ {\begin{array}{*{20}{c}} { - div({g^2}(u)\nabla u) + g(u)g'(u){{\left| {\nabla u} \right|}^2} + V(x)u = K(x)f(u) + \lambda \xi (x)g(u){{\left| {G(u)} \right|}^{p - 2}}G(u), x \in {\mathbb{R}^N},} \\ {u \in {\mathcal{D}^{1,2}}({\mathbb{R}^N}),} \end{array}} \right.$$ where N ≥ 3, V, K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Using a change of variable as $$G(u) = \int_0^u {g(t)\rm{dt}}$$, the above quasilinear equation is reduced to a semilinear one. Under some suitable assumptions, we prove that the above equation has at least one nontrivial solution by working in weighted Sobolev spaces and employing the variational methods.