Existence of ground state solution for quasilinear Schrödinger system Provisionally Accepted
Jing Zhang1*
- 1Inner Mongolia Normal University, China
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.
Keywords: Quasilinear Schrödinger system, Pohožaev identity, Ground state solution, critical point theorem, Lebesgue dominated convergence theorem
Received: 14 Feb 2024;
Accepted: 26 Mar 2024.
Copyright: © 2024 Zhang and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
* Correspondence: Mrs. Jing Zhang, Inner Mongolia Normal University, Hohhot, 130012, Inner Mongolia, China