Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition

Abstract

In this paper, we study the following quasilinear Schrödinger equation

$$-\Delta u+u-\Delta \left(u^2\right)u=h\left(u\right),x\in {\mathbb{R}}^N,$$

where $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$, $h$ is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^4}=\infty$, we only need to assume that ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^2}=\infty$.