Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction

TL;DR

This paper investigates the existence, concentration, and symmetry properties of positive and nodal solutions to a Schrödinger equation with a shrinking core, revealing limiting profiles that solve a nonautonomous elliptic equation with sign-changing nonlinearity.

Contribution

It introduces a novel analysis of solution concentration in shrinking regions and characterizes the symmetry and decay of limiting profiles in a nonautonomous setting.

Findings

Limiting profiles exhibit specific symmetries such as radial or foliated Schwarz.

Profiles decay polynomially at infinity.

Existence of solutions concentrates around the shrinking core.

Abstract

We study the existence and concentration of positive and nodal solutions to a Schr\"odinger equation in the presence of a shrinking self-focusing core of arbitrary shape. Via a suitable rescaling, the concentration gives rise to a limiting profile that solves a nonautonomous elliptic semilinear equation with a sharp sign change in the nonlinearity. We characterize the (radial or foliated Schwarz) symmetries and the (polynomial) decay of the least-energy positive and nodal limiting profiles.