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.

我们研究具有临界指数增长的规范非线性Schrödinger方程的最小能量符号转换解的存在性和渐近行为 $\begin{array}{rl}& -\mathrm{\Delta }ü+\omega ü+\lambda \left(\frac{H^2（|X|）}{|X{|}^2}+{\int }_{|X|}^{\mathrm{\infty }}\frac{H（s）}{s}{ü}^2（s）\mathrm{d}s\right)ü=F（ü）\mathrm{一世}\mathrm{ñ}{\mathbb{[R}}^2，\\ & ü\in H_{\mathrm{[R}}^{\mathrm{1个}}（{\mathbb{[R}}^2），\end{array}$ 哪里 $\omega ，\lambda >0$ 是常数， $H（s）={\int }_0^s\frac{\mathrm{[R}}{2}{ü}^2（\mathrm{[R}）\mathrm{d}\mathrm{[R}。$ 在一些适当的假设下 $F\in C（\mathbb{[R}）$，我们应用约束最小化参数来建立最小能量符号转换解决方案 ${ü}_{\lambda }$恰好有两个节点域。此外，我们证明了${ü}_{\lambda }$ 严格大于基态能量的两倍，并分析其渐近行为 ${ü}_{\lambda }$ 如 $\lambda \searrow 0^{+}$。我们的结果概括了现有的结果，请参见Li G.等。（对规范的非线性Schrödinger方程的符号转换解。JMath Anal Appl。2017; 455：1559-1578）和Liu Z.等人。非线性规范薛定ding方程中变号驻波的存在性和多重性${\mathbb{[R}}^2$。非线性。例如，2019; 32：3082–3111），到具有临界指数增长的规范非线性Schrödinger方程。