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

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