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It is known that the De Giorgi's conjecture does not hold in two dimensionsfor semilinear elliptic equations with a nonzero drift, in general, $$ \Deltau+ q\cdot \nabla u+f(u)=0 \ \ \text{in } \ \ \mathbb R^2, $$ 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 premixedflame. In this article, we prove De Giorgi type results, and stabilityconjecture, 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 prioriestimates for the above equation, when $n\ge 1$, with various jumping kernels.The operator $\Delta+cL$ is an infinitesimal generator of jump-diffusionprocesses in the context of probability theory.

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