Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in ${\mathbb{R}}^N$

Abstract

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schrödinger} equations involving Stein-Weiss type convolution \begin{align*} -Δ_N u -Δ_N (u^{2})u +V(x)|u|^{N-2}u= \left(\int_{\mathbb R^N}\frac{F(y,u)}{|y|^β|x-y|^μ}~dy\right)\frac{f(x,u)}{|x|^β} \;\; \text{ in}\; \mathbb R^N, \end{align*} where $N\ge 2,$ $0<\mu <N,\beta \ge 0,$ and $2\beta +\mu \le N.$ The potential $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous function satisfying $0<V_0\le V(x)$ for all $x\in {\mathbb{R}}^N$ and some appropriate assumptions. The nonlinearity $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)={\int }_0^{s}f(x,t)dt$ is the primitive of $f$.

arXiv commentSome mistakes and typos are corrected and new results are added in this updated version