Ground state solutions for modified quasilinear Schrödinger equations coupled with the Chern–Simons gauge theory

Abstract

In this paper, we study the following modified quasilinear Schrödinger equation $\begin{array}{rl}& -\mathrm{\Delta }u+V\left(x\right)u-\kappa u\mathrm{\Delta }\left(u^2\right)+q\frac{h^2\left(|x|\right)}{|x{|}^2}(1+\kappa u^2)u\\ & +q\left({\int }_{|x|}^{+\mathrm{\infty }}\frac{h\left(s\right)}{s}(2+\kappa u^2(s\left)\right)u^2\left(s\right)\mathrm{d}s\right)u=|u{|}^{p-2}u\mathrm{i}\mathrm{n}{\mathbb{R}}^2,\end{array}$ where $\kappa >0$, q>0, p>6 and $V\in \mathcal{C}({\mathbb{R}}^2,\mathbb{R})$. We develop a more direct and simpler approach to prove the existence of ground state solutions. Our method is based on the Poho$\check{z}$aev type identity and monotone trick.

Keywords:

• Gauged Schrödinger equation
• Poho$\check{z}$aev identity
• ground state solutions

MR(2010) Subject Classifications:

• 35J60
• 35J20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant numbers 11361042, 11771198, 11901276, 11961045] and Jiangxi Provincial Natural Science Foundation [20181BAB201003, 20202BAB201001, 20202BAB211004].