Existence of positive ground state solutions for quasilinear Schrödinger system with positive parameter

Abstract

This paper is concerned with the following quasilinear Schrödinger system in the entire space ${\mathbb{R}}^N(N\ge 3)$: $\{\begin{array}{l}-\mathrm{\Delta }u+u+{\displaystyle \frac{k}{2}}\mathrm{△}\left(u^2\right)u={\displaystyle \frac{2\alpha }{\alpha +\text{β}}}|u{|}^{\alpha -2}u|v{|}^{\text{β}},\\ -\mathrm{\Delta }v+v+{\displaystyle \frac{k}{2}}\mathrm{△}\left(v^2\right)v={\displaystyle \frac{2\text{β}}{\alpha +\text{β}}}|u{|}^{\alpha }|v{|}^{\text{β}-2}v,\end{array}$where $\alpha ,\text{β}>1$, $2<\alpha +\text{β}<2^{\ast }:=\frac{2N}{N-2}$, k>0 is a parameter. The main strategy is to define a Pohožaev manifold via analyzing the term ${\int }_{{\mathbb{R}}^N}|u{|}^{\alpha }|v{|}^{\text{β}}$, by using Moser iteration, we obtain the existence of positive ground state solutions. Our main contribution is related to the fact that we are able to deal with the case k>0 for quasilinear system.

Keywords:

• Quasilinear Schrödinger system
• positive ground state solution
• Pohožaev manifold
• Moser iteration

Mathematics Subject Classifications:

• 35J20
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11871152] and Key Project of Natural Science Foundation of Fujian [grant number 2020J02035].