Existence of a ground-state solution for a quasilinear Schrödinger system - Zhang, Xue; Zhang, Jing (Frontiers in Physics, 2024)
In this paper, we consider the following quasilinear Schr\"odinger system $$ \left\{ \begin{array}{lc} -\Delta u+ u+\frac{k}{2}[\Delta |u|^2]u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^\beta, & x\in \,\, \mathbb{R}^{N}, \\ -\Delta v+ v+\frac{k}{2}[\Delta|v|^2]v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in \,\, \mathbb{R}^{N}, \\ \end{array} \right. $$ where $k<0$ is a real constant, $\alpha >1$, $\beta >1$, $\alpha+\beta<2^*$. We take advantage of the critical point theorem developed by Jeanjean \cite{J}, combine with Poho\v{z}aev identity to obtain the existence of a ground state solution, which is the nontrivial solution with least possible energy.