Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 56, pp. 1-17.

Title: Positive solutions for asymptotically 3-linear quasilinear Schrodinger equations
       
Authors: Guofa Li, Bitao Cheng, Yisheng Huang

Abstract:
 In this article, we study the quasilinear Schrodinger equation
 $$
 -\Delta u+V(x)u-\frac{\kappa}{2}[\Delta(1+u^2)^{1/2}]\frac{u}{(1+u^2)^{1/2}}
 =h(u),\quad x\in\mathbb{R}^N,
 $$
 where $N\geq3$, $\kappa>0$ is a parameter, $V: \mathbb{R}^N\to\mathbb{R}$ is
 a given potential. The nonlinearity $h\in C(\mathbb{R}, \mathbb{R})$ is
 asymptotically 3-linear at infinity. We obtain the nonexistence of a least energy
 solution and the existence of a positive solution, via the Pohozaev manifold
 and a linking theorem. Our results improve recent results in [4,22].
 
Submitted June 16, 2019. Published June 04, 2020.
Math Subject Classifications: 35J20, 35J62.
Key Words: Quasilinear Schrodinger equations; asymptotically 3-linear;
           Pohozaev identity; linking theorem; positive solution.