Asymptotic behaviour of the least energy solutions of fractional semilinear Neumann problem

We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem:d(-?)su + u = |u|p-1 u in ?, Nsu = 0 in Rn \ ?, u > 0 in ?,where ? ? Rn is a bounded domain of class C1,1, 1 < p < n+s/n?s, n > max {1, 2s} , 0 < s < 1, d > 0 and Nsu is the nonlocal Neumann derivative. We show that for small d, the least energy solutions ud of the above problem achieves L? bound independent of d. Using this together with suitable Lr-estimates on ud, we show that least energy solution ud achieve maximum on the boundary of ? for d sufficiently small.