Positive ground state solutions for the Chern–Simons–Schrödinger system

Abstract

In this paper we investigate the following nonlinear Chern–Simons–Schrödinger system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+ V(|x|)u+\lambda \left( \int ^\infty _{|x|}\frac{h(s)}{s}u^2(s)ds+\frac{h^2(|x|)}{|x|^2}\right) u=|u|^{p-2}u~~ \text {in}~ {\mathbb {R}}^2,\\ u(x)=u(|x|)\in H^1({\mathbb {R}}^2), \end{array}\right. } \end{aligned}$$

where $\lambda >0$, the Chern–Simons term $h(s)=\int _0^s\frac{r}{2}u^2(r)dr=\frac{1}{4\pi }\int _{B_s}u^2(x)dx$ and V(|x|) is an external potential. Under some suitable conditions on the external potential, we prove the existence of a positive ground state solution for $p\in (6,+\infty )$ via the Pohožaev–Nehari manifold method and the global compactness lemma. The novelty of this works with respect to some recent results is that we establish a key lemma which is analogous to the Br$\acute{e}$zis–Lieb convergence lemma. Furthermore, we also prove concentration of these ground state solutions. As its supplementary results, we give the several nonexistence results of nontrivial solutions.