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

Abstract

We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem:d(-∆)su + u = |u|p-1 u in Ω, Nsu = 0 in Rn \ Ω, u > 0 in Ω,where Ω ⊂ 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 Ω for d sufficiently small.