Semilinear elliptic equations involving mixed local and nonlocal operators

TL;DR

This paper studies elliptic equations combining local and nonlocal operators, proving symmetry properties of solutions using the moving plane method, with implications for classical symmetry conjectures.

Contribution

It introduces a framework for analyzing mixed local and nonlocal elliptic operators and establishes symmetry results for their solutions.

Findings

Proves radial symmetry of solutions using the moving plane method.

Establishes one-dimensional symmetry related to Gibbons' conjecture.

Demonstrates the applicability of symmetry techniques to combined operators.

Abstract

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form $-\Delta+(-\Delta)^s$, with $s\in(0,1)$. We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.