Existence, Nonexistence and Multiplicity Results of a Chern-Simons-Schrödinger System

We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$ where $\lambda >0$ is a parameter, $p\in (2,4)$ and $$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$ We prove that the system has no solutions for $\lambda $ large and has two radial solutions for $\lambda $ small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of $\lambda $. Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).