Nonlocal Harnack inequalities for nonlocal Schrödinger operators with A1-Muckenhoupt potentials

Abstract

In this paper, by applying the De Giorgi-Nash-Moser theory we obtain nonlocal Harnack inequalities for (locally nonnegative in Ω) weak solutions of the nonlocal Schrödinger equations{LKu+Vu=0 in Ω,u=g in Rn∖Ω where V=V+−V− with V−∈Lloc1(Rn) and V+∈Llocq(Rn) (q>n2s>1, 0<s<1) is a potential such that (V−,V+b,k) belongs to the (A1,A1)-Muckenhoupt class and V+b,k is in the A1-Muckenhoupt class for all k∈N (here, V+b,k:=V+max⁡{b,1/k}/b for a nonnegative bounded function b on Rn with V+/b∈Llocq(Rn)), LK is an integro-differential operator, Ω⊂Rn is a bounded domain with Lipschitz boundary and g∈Hs(Rn). Interestingly, this result implies the classical Harnack inequalities for globally nonnegative weak solutions of the equations. In addition, we obtain nonlocal weak Harnack inequalities of its weak supersolutions. In particular, we note that all the above results are still working for any nonnegative potential in Llocq(Rn) (q>n2s>1, 0<s<1).