Existence of positive solutions for a class of quasilinear Schr\"{o}dinger equations with critical Choquard nonlinearity

TL;DR

This paper proves the existence of positive solutions for a class of quasilinear Schrödinger equations with critical Choquard nonlinearity using variational methods and change of variables.

Contribution

It introduces new existence results for positive solutions to a complex quasilinear Schrödinger Choquard equation with critical nonlinearity, under specific assumptions.

Findings

Established existence of positive weak solutions

Applied variational methods and change of variables

Handled critical Choquard nonlinearity in unbounded domain

Abstract

This article is concerned with the existence of positive weak solutions for the following quasilinear Schr\"odinger 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 $N \geq 3$, $\displaystyle k(x,u) := h(x,u) + (I_{\vartheta}*|u|^{\alpha\cdot2^*_\mu})|u|^{\alpha\cdot2^*_\mu-2}u$, $g : \mathbb{R} \to \mathbb{R}^+$ is a differentiable even function with $g(0) = 1$ and $g'(t) \geq 0$ for all $t \geq 0$; $h\in C( \mathbb{R}^N \times\mathbb{R}, \mathbb{R})$ 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 $a$.