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]. For more information see https://ejde.math.txstate.edu/Volumes/2020/56/abstr.html