Qualitative properties of positive solutions for mixed integro-differential equations

TL;DR

This paper investigates the decay and symmetry of positive solutions to a mixed integro-differential equation involving fractional Laplacian and Laplacian operators, addressing challenges posed by their combined nonlocal and local nature.

Contribution

It introduces new decay and symmetry results for solutions of mixed integro-differential equations, utilizing a novel Hopf's Lemma and the moving planes method.

Findings

Solutions decay to zero at infinity.

Solutions exhibit symmetry properties.

A new Hopf's Lemma is established.

Abstract

This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation \begin{equation}\label{eq 1} \left\{ \arraycolsep=1pt \begin{array}{lll} (-\Delta)_x^{\alpha} u+(-\Delta)_y u+u=f(u)\quad \ \ {\rm in}\ \ \R^N\times\R^M, u>0\ \ {\rm{in}}\ \R^N\times\R^M,\ \ \quad \lim_{|(x,y)|\to+\infty}u(x,y)=0, \end{array} \right. \end{equation} with $N\ge 1$, $M\ge 1$ and $\alpha\in (0,1)$. We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.