AUTHOR=Zhang Xue , Zhang Jing 

TITLE=Existence of a ground-state solution for a quasilinear Schrödinger system

JOURNAL=Frontiers in Physics

VOLUME=Volume 12 - 2024

YEAR=2024

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2024.1386144

DOI=10.3389/fphy.2024.1386144

ISSN=2296-424X

ABSTRACT=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.