Sign-changing solutions for the nonlinear Chern–Simons–Schrödinger equations

ABSTRACT

In this paper, we study the existence and asymptotic behavior of least energy sign-changing solutions for the nonlinear Chern–Simons–Schrödinger equations $\begin{array}{l}-\mathrm{\Delta }u+\omega u+\lambda \left(\frac{h^2\left(|x|\right)}{|x{|}^2}+{\int }_{|x|}^{\mathrm{\infty }}\frac{h\left(s\right)}{s}{u}^2\left(s\right)\mathrm{d}s\right)u=f\left(u\right),x\in {\mathbb{R}}^2,\\ u\in H_r^1\left({\mathbb{R}}^2\right),\end{array}$ where $\omega ,\lambda >0$, $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ and $h\left(s\right)=\frac{1}{2}{\int }_0^{s}r{u}^2\left(r\right)\mathrm{d}r.$ Under suitable assumptions on f, we use some analytical skills and constraint minimization method to show that the above problem admits one least energy sign-changing solution $u_{\lambda }$ with precisely two nodal domains. Furthermore, we show that the energy of $u_{\lambda }$ is strictly larger than two times of the least energy, and present a convergence property of $u_{\lambda }$ as $\lambda \to 0^{+}$. Finally, we also prove that the above results are valid for the Chern–Simons–Schrödinger equations with steep well potential.

KEYWORDS:

• Chern–Simons–Schrödinger equations
• asymptotic behavior
• least energy sign-changing solutions
• steep well potential

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J60
• 35J20

COMMUNICATED BY:

• Junping Shi

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Weihong Xie http://orcid.org/0000-0002-1762-988X

Additional information

Funding

This work is supported by the Fundamental Research Funds for the Central Universities of Central South University [grant number 2018zzts006].