NONLOCAL HARNACK INEQUALITIES FOR NONLOCAL p-LAPLACIAN TYPE SCHRODINGER OPERATORS WITH <i>A</i>₁<i><SUP>p</SUP></i>-MUCKENHOUPT POTENTIALS

초록

In this paper, we obtain nonlocal Harnack inequalities for (locally nonnegative in S2) weak solutions of the nonlocal p-Laplacian type Schrodinger equation LpKu + V|u|p-2u = 0 in S2, u = g in Rn \ S2 where V = V+ - V- with V- E L1loc(Rn) and V+ E Lqloc(Rn) for q > psn> 1 ( p > 1, s E (0, 1)) is a potential such that (V-, V b,i + ) belongs to the (A1, A1)-Muckenhoupt class and V b,i + is in the A1-Muckenhoupt class for all i E N (here, V b,i + := V+ max{b, 1/i} for an almost everywhere positive bounded function b on Rn with V+/b E Lqloc(Rn)), LpK is an integro-differential operator of p Laplacian type, S2 C Rn is a bounded domain with Lipschitz boundary and g E W s,p(Rn). In addition, we obtain nonlocal weak Harnack inequalities of its weak supersolutions.In particular, all the above results are still working for any nonnegative potential in Lqlo c(Rn) (q > psn> 1, p > 1, 0 < s < 1).

키워드