Threshold for Existence, Non-existence and Multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ball

TL;DR

This paper investigates the existence, non-existence, and multiplicity of normalized solutions for a nonlinear Schrödinger equation outside a ball, revealing how domain shape and prescribed mass influence solution behavior, including stability properties.

Contribution

It provides the first threshold results for solutions in an exterior domain, highlighting the impact of prescribed mass and domain shape on solution multiplicity and stability, with new critical exponents identified.

Findings

Prescribed mass affects the number of solutions.

Solutions are stable in exterior domains, unlike in full space.

A new critical exponent p=6 for N=2 is identified.

Abstract

We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surprising" and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in RN . In this paper, we will show that in the exterior domain they are strongly stable.