Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction

Abstract

This paper is concerned with the following quasilinear Schrödinger equation:

where $N\ge 3$, $V\in \mathcal{C}({\mathbb{R}}^N,[0,\infty ))$ and $g\in \mathcal{C}(\mathbb{R},\mathbb{R})$ is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under mild assumptions on $V$ and $g$. Moreover, we establish a minimax characterization of the ground state energy. Especially, we impose some new conditions on $V$ and more general assumptions on $g$. For this, some new tricks are introduced to overcome the competing effect between the quasilinear term and the superlinear reaction. Hence our results improve and extend recent theorems in several directions.