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

### Sitong Chen

Central South University, Changsha, Hunan, China### Vicenţiu D. Rădulescu

AGH University of Science and Technlogy, Kraków, Poland and Romanian Academy, Bucharest, Romania### Xianhua Tang

Central South University, Changsha, Hunan, China### Binlin Zhang

Shandong University of Science and Technology, Qingdao, and Heilongjiang Institute of Technology, Harbin, China

## 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.

## Cite this article

Sitong Chen, Vicenţiu D. Rădulescu, Xianhua Tang, Binlin Zhang, Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction. Rev. Mat. Iberoam. 36 (2020), no. 5, pp. 1549–1570

DOI 10.4171/RMI/1175