Positive solutions for asymptotically 3-linear quasilinear Schrodinger equations

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].

Keywords

• quasilinear schrodinger equations
• asymptotically 3-linear
• pohozaev identity
• linking theorem
• positive solution