Sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth

Abstract

We study the existence and asymptotic behavior of least energy sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth $\begin{array}{rl}& -\mathrm{\Delta }u+\omega u+\lambda \left(\frac{h^2\left(|x|\right)}{|x{|}^2}+{\int }_{|x|}^{\mathrm{\infty }}\frac{h\left(s\right)}{s}{u}^2\left(s\right)\mathrm{d}s\right)u=f\left(u\right)\mathrm{i}\mathrm{n}{\mathbb{R}}^2,\\ & u\in H_r^1\left({\mathbb{R}}^2\right),\end{array}$ where $\omega ,\lambda >0$ are constants and $h\left(s\right)={\int }_0^s\frac{r}{2}{u}^2\left(r\right)\mathrm{d}r.$ Under some suitable assumptions on $f\in C\left(\mathbb{R}\right)$, we apply the constraint minimization argument to establish a least energy sign-changing solution $u_{\lambda }$ with precisely two nodal domains. Moreover, we show that the energy of $u_{\lambda }$ is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of $u_{\lambda }$ as $\lambda \searrow 0^{+}$. Our results generalize the existing ones, see Li G. et al. (Sign-changing solutions to a gauged nonlinear Schrödinger equation. J Math Anal Appl. 2017;455:1559–1578) and Liu Z. et al. (Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in ${\mathbb{R}}^2$. Nonlinearity. 2019;32:3082–3111) for example, to the gauged nonlinear Schrödinger equation with critical exponential growth.

Keywords:

• Least energy sign-changing solutions
• Chern–Simons gauge theory
• ground state energy
• asymptotic
• critical exponential growth

AMS Subject Classifications:

• 35J20
• 35Q55

Acknowledgments

The author would like to express his sincere and warmest thanks to the anonymous referees for carefully reading the manuscript and valuable comments and suggestions.

Disclosure statement

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

Additional information

Funding

The author was supported by the fundamental Research Funds for the Central Universities (WUT: 2019IVA107 and 2020IB019).