Abstract:

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