HOLDER REGULARITY OF WEAK SOLUTIONS TO NONLOCAL p-LAPLACIAN TYPE SCHRODINGER EQUATIONS WITH A<SUP>p</SUP>1-MUCKENHOUPT POTENTIALS

초록

In this article, using the De Giorgi-Nash-Moser method, we obtain an interior Holder continuity of weak solutions to nonlo cal p-Laplacian type Schrodinger equations given by an integro-differential operator L-K(p) (p > 1), L(K)(p)u +V|u|(p-2)u = 0 in ohm, u = g in R-n\ohm. Where V = V+ - V(-)with (V-, V+) is an element of L-loc(1)(R-n) x L-loc(q)(R-n) for q > n/ps > 1 and 0 < s < 1 is a potential such that (V-, V-+(b,i) ) belongs to the (A(1), A(1))-Muckenhoupt class and V-b,V-i (+) is in the A(1)-Muckenhoupt class for all i is an element of N. Here, V-+(b,i) := V+ max{b, 1/i}/b for an almost everywhere positive bounded function b on R-n with V+/b is an element of L-loc(q)(R-n), g is an element of W-s,W-p(R-n) and ohm subset of R-n is a bounded domain with Lipschitz boundary. In addition, we prove local boundedness of weak subsolutions of the nonlocal p-Laplacian type Schrodinger equations. Also we obtain the logarithmic estimate of the weak supersolutions which play a crucial role in proving Holder regularity of the weak solutions. We note that all the above results also work for a nonnegative potential in L-loc(q)(R-n) (q > n/ps > 1, 0 < s < 1).

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