Strong Comparison Principle for the Fractional p-Laplacian and Applications to Starshaped Rings

In the following, we show the strong comparison principle for the fractional p-Laplacian, i.e. we analyze {(-Delta)(p)(s)v + q(x)vertical bar v vertical bar(p-2) v >= 0 in D, (-Delta(s)(p)w + q(x)vertical bar w vertical bar(p-2) w <= 0 in D, v >= w in R-N, where s is an element of (0, 1), p > 1, D subset of R-N is an open set, and q is an element of L-infinity (R-N) is a nonnegative function. Under suitable conditions on s, p and some regularity assumptions on v, w, we show that either v w in R-N or v > w in D. Moreover, we apply this result to analyze the geometry of nonnegative solutions in starshaped rings and in the half space.