Ground state solutions for the Chern–Simons–Schrödinger equations with general nonlinearity

ABSTRACT

We consider the nonlinear Chern–Simons–Schrödinger equations with general nonlinearity $-\mathrm{\Delta }u+wu+\left({\int }_{|x|}^{\mathrm{\infty }}\frac{h\left(s\right)}{s}{u}^2\left(s\right)\mathrm{d}s\right)u+\frac{h^2\left(|x|\right)}{|x{|}^2}u=f\left(u\right),x\in {\mathbb{R}}^2,$ where $h\left(s\right)={\int }_0^s\left(r/2\right)u^2\left(r\right)\mathrm{d}r=\left(1/4\pi \right){\int }_{B_s}{u}^2\left(x\right)\mathrm{d}x$ and w>0. With the condition $\underset{|t|\to \mathrm{\infty }}{lim}F\left(t\right)/\left(|t{|}^{6-2/\alpha }\right)=\mathrm{\infty }$, we obtain the ground state solution of the Nehari–Pohozaev type. Based on the result, by the Jeanjeans monotonicity trick, we also get a least energy solution. For the case $\underset{|t|\to \mathrm{\infty }}{lim}\left(F\left(t\right)/t^6\right)=\mathrm{\infty }$, the existence of ground state solution is proved by the diagonal method. We generalize the existence results.

KEYWORDS:

• Chern–Simons–Schrödinger equations
• ground state solution
• Nehari–Pohozaev type
• diagonal method

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 58E50

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ning Zhang  http://orcid.org/0000-0003-3193-1054

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11571370). This work was also supported by the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 502211908).