Holder Continuity of Weak Solutions to Nonlocal Schrodinger Equations with A(1)-Muckenhoupt Potentials

Abstract

By applying the De Giorgi-Nash-Moser theory, we obtain an interior Holder continuity of weak solutions to nonlocal Schrodinger equations given by an integro-differential operator L-K as follows; {L(K)u + Vu = 0 in Omega, u = g inR(n)\Omega where V = V+ - V- with V- is an element of L-loc(1) (R-n) and V+ is an element of L-loc(q)(R-n) (q > n/2s > 1,0 < s <1) is a potential such that (V-, V-+(b,k)) belongs to the (A(1), A(1))- Muckenhoupt class and V-+(b,k) is in the A(1)-Muckenhoupt class for all k is an element of N (here V-+(b,k) = V+ max{b, 1/k}/b for a nonnegative bounded function b on R-n with V+/b is an element of L-loc(q)(R-n)), g is an element of H-s (R-n) and Omega is a bounded domain in R-n with Lipschitz boundary. In addition, we get the local boundedness of weak subsolutions of the nonlocal Schrodinger equations. In particular, we note that all the above results still work for any nonnegative potential in L-loc(q)(Rn) (q > n/2s > 1,0 < s < 1).