We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system
{−Δu+u+λ(h2(|x|)|x|2+∫|x|+∞h(s)su2(s)ds)u=|u|p−2u,x∈R2,u∈Hr1(R2),\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$ \end{document} where λ>0\documentclass[12pt]{minimal}
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\begin{document}$\lambda >0$\end{document} is a parameter, p∈(2,4)\documentclass[12pt]{minimal}
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\begin{document}$p\in (2,4)$\end{document} and
h(s)=12∫0sru2(r)dr.\documentclass[12pt]{minimal}
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\begin{document}$$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$\end{document} We prove that the system has no solutions for λ\documentclass[12pt]{minimal}
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\begin{document}$\lambda $\end{document} large and has two radial solutions for λ\documentclass[12pt]{minimal}
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\begin{document}$\lambda $\end{document} small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of λ\documentclass[12pt]{minimal}
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\begin{document}$\lambda $\end{document}. Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).
