Some rigidity results and asymptotic porperties for solutions to semilinear elliptic P.D.E

TL;DR

This paper investigates rigidity and asymptotic properties of solutions to semilinear elliptic PDEs, focusing on symmetry and Liouville-type theorems, extending known results related to the Cahn-Hilliard equation.

Contribution

It introduces new rigidity results and symmetry theorems for solutions of semilinear elliptic equations with general potentials, broadening understanding beyond classical cases.

Findings

Established new Liouville-type theorems for solutions.

Proved symmetry results under general potential conditions.

Extended known properties from specific models like Cahn-Hilliard to broader classes.

Abstract

We will present some rigidity results for solutions to semilinear elliptic equations of the form $\Deltau = W'(u)$, where W is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouvlle-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.