Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in ${\mathbb{R}}^2$

Abstract

We are concerned with sign-changing solutions of the following gauged nonlinear Schrödinger equation in dimension two including the so-called Chern-Simons term \begin{align*} \left\{ \begin{array}{ll} -\triangle {u}+ωu+\left(\frac{h^2(|x|)}{|x|^2}+\int_{|x|}^{+\infty}\frac{h(s)}{s}u^2(s){\rm ds}\right) u =λ|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 ${\lambda }_n\to \infty$ as $n\to \infty$, up to a subsequence, $λ_n^{\frac{1}{p-2}}w_{λ_n}\rg w$ strongly in $H_{rad}^1({\mathbb{R}}^2)$ as $n\to \infty$, where $w$ is a sign-changing solution of $$-△u+\omega u=|u{|}^{p-2}u,u\in H_{rad}^1({\mathbb{R}}^2).$$

28 pages