# Revista Matemática Iberoamericana

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**Published online first: 2020-02-10**

**DOI: 10.4171/rmi/1175**

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

Sitong Chen^{[1]}, Vicenţiu D. Rădulescu

^{[2]}, Xianhua Tang

^{[3]}and Binlin Zhang

^{[4]}(1) Central South University, Changsha, Hunan, China

(2) AGH University of Science and Technlogy, Kraków, Poland and Romanian Academy, Bucharest, Romania

(3) Central South University, Changsha, Hunan, China

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

This paper is concerned with the following quasilinear Schrödinger equation: $$-\Delta u+V(x)u-\frac{1}{2}\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N,$$ 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.

*Keywords: *Quasilinear Schrödinger equation, ground state solution, Pohozaev identity

Chen Sitong, Rădulescu Vicenţiu, Tang Xianhua, Zhang Binlin: Ground state solutions for quasilinear Schrödinger equations with variable potential and superlinear reaction. *Rev. Mat. Iberoam.* Electronically published on February 10, 2020. doi: 10.4171/rmi/1175 (to appear in print)