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

We consider the minimizers of L²-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain Ω of RN which contains $0$. We prove that there is a threshold a*> 0 such that minimizers exist for 0< a< a* and the minimizer does not exist for any a> a*. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold a^* 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 a^*, we prove local uniqueness of minimizers when a is close enough to a^*.