On nonlocal systems with jump processes of finite range and with decays

We study the following system of equations $$L_i(u_i) = H_i(u_1,\cdots,u_m) \quad \text{in} \ \ \mathbb R^n ,$$ when $m\ge 1$, $u_i: \mathbb R^n \to \mathbb R$ and $H=(H_i)_{i=1}^m$ is a sequence of general nonlinearities. The nonlocal operator $L_i$ is given by $$L_i(f (x)):= \lim_{\epsilon\to 0} \int_{\mathbb R^n \setminus B_\epsilon(x) } [f(x) - f(z)] J_i(z-x) dz,$$ for a sequence of even, nonnegative and measurable jump kernels $J_i$. We prove a Poincar\'{e} inequality for stable solutions of the above system for a general jump kernel $J_i$. In particular, for the case of scalar equations, that is when $m=1$, it reads \begin{equation*}\label{} \iint_{ \mathbb R^{2n}} \mathcal A_y(\nabla_x u) [\eta^2(x)+\eta^2(x+y)] J(y) dx dy \le \iint_{ \mathbb R^{2n}} \mathcal B_y(\nabla_x u) [ \eta(x) - \eta(x+y) ] ^2 J(y) d x dy , \end{equation*} for any $\eta \in C_c^1(\mathbb R^{n})$ and for some nonnegative $\mathcal A_y(\nabla_x u)$ and $\mathcal B_y(\nabla_x u)$. This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in \cite{sz} for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever $H_i(u)\ge 0$ or $\sum_{i=1}^m u_i H_i(u)\le 0$ then Liouville theorems hold for each $u_i$ in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel $J_i$ and a Liouville theorem for the quotient of partial derivatives of $u$.