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

TL;DR

This paper studies the existence, nonexistence, and uniqueness of minimizers for a class of inhomogeneous variational problems with spatially decaying nonlinearities, revealing how the potential influences these properties.

Contribution

It establishes the existence threshold for minimizers, analyzes their concentration behavior near the threshold, and demonstrates local uniqueness under certain conditions.

Findings

Existence of minimizers for 0 < a < a*

Nonexistence of minimizers for a > a*

Local uniqueness of minimizers near the threshold a*

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 $a^{*}>0$ 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 $a\nearrow a^*$, we prove local uniqueness of minimizers when $a$ is close enough to $a^*$.