A rigidity result for nonlocal semilinear equations

TL;DR

This paper proves that solutions to certain anisotropic nonlocal semilinear equations must be constant if they grow slower than the operator’s order at infinity.

Contribution

It establishes a rigidity result for nonlocal semilinear equations, showing solutions are constant under specific growth conditions.

Findings

Solutions with sub-operator growth are constant

The result applies to anisotropic integro-differential operators

Provides a new criterion for solution constancy in nonlocal equations

Abstract

We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.