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

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Abstract

We study the following system of equationsLi(ui)=Hi(u1,,um)inRn, when m1, ui:RnR and H=(Hi)i=1m is a sequence of general nonlinearities. The nonlocal operator Li is given byLi(f(x)):=limϵ0RnBϵ(x)[f(x)f(z)]Ji(zx)dz, for a sequence of even, nonnegative and measurable jump kernels Ji. We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel Ji. In particular, for the case of scalar equations, that is when m=1, it readsR2nAy(xu)[η2(x)+η2(x+y)]J(y)dxdyR2nBy(xu)[η(x)η(x+y)]2J(y)dxdy, for any ηCc1(Rn) and for some nonnegative Ay(xu) and By(xu). This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in [46] 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 Hi(u)0 or i=1muiHi(u)0 then Liouville theorems hold for each ui in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel Ji and a Liouville theorem for the quotient of partial derivatives of u.

MSC

35J60
60J75
60J35
35B35
35B32

Keywords

Nonlocal equations
De Giorgi's conjecture
Stable solutions
Integral inequalities
Energy estimates

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