De Giorgi type results for equations with nonlocal lower-order terms

Abstract

It is known that the De Giorgi’s conjecture does not hold in two dimensions for semilinear elliptic equations with a nonzero drift, in general, $$\begin{aligned} \Delta u+ q\cdot \nabla u+f(u)=0 \ \ \text {in } \ \ {\mathbb {R}}^2, \end{aligned}$$when \(q=(0,-c)\) for \(c\ne 0\). This equation arises in the modeling of Bunsen burner flames. Bunsen flames are usually made of two flames: a diffusion flame and a premixed flame. In this article, we prove De Giorgi type results, and stability conjecture, for the following local-nonlocal counterpart of the above equation (with a nonlocal premixed flame) in two dimensions, $$\begin{aligned} \Delta u + c L[u] + f(u)=0 \quad \text {in} \ \ {{\mathbb {R}}}^n, \end{aligned}$$when L is a nonlocal operator, \(f\in C^1({\mathbb {R}})\) and \(c\in {\mathbb {R}}^+\). In addition, we provide a priori estimates for the above equation, when \(n\ge 1\), with various jumping kernels. The operator \(\Delta +cL\) is an infinitesimal generator of jump-diffusion processes in the context of probability theory.