Minimizers of $L^2$-critical inhomogeneous variational problems with a spatially decaying nonlinearity in bounded domains

Abstract

We consider the minimizers of $L^2$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $\Omega$ of ${\mathbb{R}}^N$ which contains $0$. We prove that there is a threshold $a^{\ast }>0$ such that minimizers exist for $0<a<a^{\ast }$ and the minimizer does not exist for any $a>a^{\ast }$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^{\ast }$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, based on a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^{\ast }$, we prove local uniqueness of minimizers when $a$ is close enough to $a^{\ast }$.