Existence of positive solutions for nonlinear fractional Neumann elliptic equations

This article is devoted to study the fractional Neumann elliptic problem ⎧⎪⎨ ⎪⎩ ε2s(−Δ)su+u = up in Ω, ∂νu = 0 on ∂Ω, u > 0 in Ω, where Ω is a smooth bounded domain of RN , N > 2s , 0 < s s0 < 1 , 1 < p < (N +2s)/(N− 2s) , ε > 0 and ν is the outer normal to ∂Ω . We show that there exists at least one nonconstant solution uε to this problem provided ε is small. Moreover, we prove that uε ∈ L∞(Ω) by using Moser-Nash iteration.