Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schr\"{o}dinger equation in $\R^2$

Abstract

We are concerned with sign-changing solutions of the following gauged nonlinear Schr\"{o}dinger equation in dimension two including the so-called Chern-Simons term \begin{align*} \left\{ \begin{array}{ll} -\triangle {u}+\omega u+\left(\frac{h^2(|x|)}{|x|^2}+\int_{|x|}^{+\infty}\frac{h(s)}{s}u^2(s){\rm ds}\right) u =\lambda|u|^{p-2}u& \mbox{in}\,\,\R^2, u(x)=u(|x|)\, \in\, H^1(\R^2), \end{array} \right. \end{align*} where $\omega,\lambda>0$, $p\in(4,6)$ and $$ h(s)=\frac{1}{2}\int_0^s\tau u^2(\tau)d\tau. $$ Via a novel perturbation approach and the method of invariant sets of descending flow, we investigate the existence and multiplicity of sign-changing solutions. Moreover, {\it energy doubling} is established, i.e., the energy of sign-changing solution $w_\lambda$ is strictly larger than twice that of the ground state energy for $\lambda>0$ large. Finally, for any sequence $\lambda_n\rightarrow\infty$ as $n\rightarrow\infty$, up to a subsequence, $\lambda_n^{\frac{1}{p-2}}w_{\lambda_n}\rg w$ strongly in $H_{rad}^1(\R^2)$ as $n\rightarrow\infty$, where $w$ is a sign-changing solution of $$ -\triangle {u}+\omega u=|u|^{p-2}u,\,\,u\in H_{rad}^1(\R^2). $$