Symmetry properties of some solutions to some semilinear elliptic equations

TL;DR

This paper establishes symmetry properties of solutions to certain semilinear elliptic equations, showing under what conditions solutions decay or are symmetric in specific variables, advancing understanding of solution structure.

Contribution

It provides new symmetry results for solutions decaying in some directions, based on conditions like periodicity and derivative decay, for equations with nonincreasing nonlinearities.

Findings

Solutions can be radially symmetric under certain decay conditions

Periodic solutions exhibit symmetry in specific variables

Derivative decay implies symmetry in solutions

Abstract

In this paper we prove some symmetry results for entire solutions to the semilinear equation $-\Delta u=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give some sufficient conditions for them to be radially symmetric with respect to those variables, such as periodicity or the pointwise decay of some derivatives.