Existence of positive solutions for a class of quasilinear Schrödinger equations with critical Choquard nonlinearity

Abstract

This article is concerned with the existence of positive weak solutions for the following quasilinear Schrödinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u = k(x, u) \;\text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $N\ge 3$, $k(x,u):=h(x,u)+(I_{\vartheta }\ast |u{|}^{\alpha \cdot 2_{\mu }^{\ast }})|u{|}^{\alpha \cdot 2_{\mu }^{\ast }-2}u$, $g:\mathbb{R}\to {\mathbb{R}}^{+}$ is a differentiable even function with $g(0)=1$ and $g^{\prime }(t)\ge 0$ for all $t\ge 0$; $h\in C({\mathbb{R}}^N\times \mathbb{R},\mathbb{R})$ and the potential $a\in C({\mathbb{R}}^N,\mathbb{R})$. We establish the existence of a positive solution using the change of variable and variational methods under appropriate assumptions on $g$, $h$ and $a$.