Sign-Changing Solutions for Chern–Simons–Schrödinger Equations with Asymptotically 5-Linear Nonlinearity

Abstract

In this paper, we study the following Chern–Simons–Schrödinger equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$

where $\omega ,\lambda >0$ and $h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r$. Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as $\lambda \rightarrow 0$.