Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for $N\geq 3$

TL;DR

This paper investigates the stability of flat ground state solutions to certain non-Lipschitz semilinear elliptic equations, showing they are unstable in low dimensions but can be stable in higher dimensions under specific conditions.

Contribution

It demonstrates the dimension-dependent stability of flat solutions in non-Lipschitz semilinear elliptic equations, highlighting new stability phenomena in higher dimensions.

Findings

Solutions are unstable for N=1,2.

Solutions can be stable for N≥3.

Stability depends on the exponents in the equations.

Abstract

We prove that flat ground state solutions ($i.e.$ minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions $N=1,2$ and they can be stable for $N\geq 3$ for suitable values of the involved exponents.