Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 716-729.

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Research on the Lowest Energy Solution ofChern-Simons-Schrödinger Equation with Trapping Potential

Ying Yang(),Liejun Shen*()   

  1. Center of Mathematics, Wuhan University of Technology, Wuhan 430070
  • Received:2021-06-03 Online:2022-06-26 Published:2022-05-09
  • Contact: Liejun Shen;
  • Supported by:
    the NSFC(11931012);the NSFC(11871387)


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 $p\nearrow2$.

Key words: Chern-Simons-Schrödinger equation, Energy estimate, Constrained minimization, Limit behavior

CLC Number: 

  • O175.2