%A Ying Yang,Liejun Shen %T Research on the Lowest Energy Solution ofChern-Simons-Schrödinger Equation with Trapping Potential %0 Journal Article %D 2022 %J Acta mathematica scientia,Series A %R %P 716-729 %V 42 %N 3 %U {http://121.43.60.238/sxwlxbA/CN/abstract/article_16693.shtml} %8 2022-06-26 %X

In this paper, we mainly study the existence of solutions with prescribed $L^{2}$-norm to the Chern-Simons-Schrödinger (CSS) equation. This type problem can be transformed into look for the minimizer of the corresponding energy functional $E^\beta_{p} (u)$ under the constraint $\|u\|_{L^{ 2}(\mathbb{R}^2)}=1$. Concerning the subcritical mass case, that is, $p\in(0,2)$, no matter whether the potential function $V(x)$ equals to $0$, we prove that the constraint minimization can be achieved by some simple methods. We are also concerned with the critical mass case of $p=2$:if $V(x)\equiv0$, there exist two constants $\beta^*>\beta_*>0$ which can be explicitly determined such that the constraint minimization cannot achieved for any $\beta\in(0,\beta_{*}]\cup(\beta^{*},+\infty)$; if $V(x)\not\equiv0$, the constraint minimization cannot be achieved for $\beta>\beta^{*}$, but can be achieved for $\beta\in(0,\beta_{*}]$. In addition, we discuss the limit behavior of the mass subcritical constrained minimum energy when $p\nearrow2$.