Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (1): 123-131.

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Limit Behavior of Mass Critical Inhomogeneous Schrödinger Equation at the Threshold

Deke Li(),Qingxuan Wang*()   

  1. College of Mathematics and Computer Science, Zhejiang Normal University, Zhejiang Jinhua 321004
  • Received:2022-03-24 Revised:2022-08-05 Online:2023-02-26 Published:2023-03-07
  • Supported by:
    The NSFC(11801519)


This paper is concerned with the following time-independent critical inhomogeneous Schr?dinger equation with attractive interactions: $\begin{eqnarray*} -\triangle u+ |x|^2u-\, a m(x)|u|^\frac{4}{N} u= \mu u, \ \ \mbox{in $\Bbb R^N$, $N\geq 1$.} \end{eqnarray*}$ where $a>0$, $0< m(x)\leq 1$. We will show the existence of ground states at the threshold $a=a^*$ for $m(x)=1-\lambda g(x)$ with $\lambda>0$ and suitable $0\leq g(x)<1$, and then investigate the limit behavior of those threshold ground states as $\lambda\rightarrow 0^+$. These conclusions extend the results of Deng-Guo-Lu[10, 11]. In particular, compared to the arguments of [10, 11], we use a direct and simpler method to obtain the lower bound of energy.

Key words: Inhomogeneous nonlinear Schr?dinger equation, Mass critical, Ground states solutions, Limit behaviors

CLC Number: 

  • O175.2