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

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, $$\Delta u+ q\cdot \nabla u+f(u)=0 \ \ \text{in } \ \ \mathbb R^2,$$ when $q=(0,-c)$ for $c\neq 0$. 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, $$\Delta u + c L[u] + f(u)=0 \quad \text{in} \ \ \mathbb R^n,$$ 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.