Existence of Positive Solutions for a class of Quasilinear Schrödinger Equations of Choquard type

Abstract

In this paper, we study the following quasilinear Schrödinger equation of Choquard type $$-△u+V(x)u-△(u^2)u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$$ where $N\ge 3$,\ $0<\alpha <N$, $\frac{2(N+\alpha )}{N}\le p<\frac{2(N+\alpha )}{N-2}$ and $I_{\alpha }$ is a Riesz potential. Under appropriate assumptions on $V(x)$, we establish the existence of positive solutions.