Minimizers of $L^2$-Subcritical Inhomogeneous Variational Problems with A Spatially Decaying Nonlinearity

TL;DR

This paper investigates the behavior of minimizers in inhomogeneous variational problems with nonlinearities that decay spatially, especially near singular points, revealing their concentration patterns as the mass parameter grows large.

Contribution

It provides a detailed analysis of the limit concentration behavior of minimizers in inhomogeneous variational problems with spatially decaying nonlinearities, including singularities at the origin.

Findings

Minimizers exhibit concentration phenomena as mass increases.

Refined analysis of nonlinear decay near singular points.

Characterization of limit behavior of minimizers.

Abstract

We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by establishing the refined analysis of the spatially decaying nonlinear term.