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: \begin{align*} \left{\begin{array}{l l} { d(-Δ)^{s}u+ u= \abs{u}^{p-1}u } $\text{ in }\mathrm{\Omega },$ { $\mathcal{N}$_{s}u=0 } $\text{ in }{\mathbb{R}}^n\setminus \bar{\mathrm{\Omega }},$ {u>0} $\text{ in }\mathrm{\Omega },$ \end{array} \right.\end{align*} where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain of class $C^{1,1}$, $1<p<\frac{n+s}{n-s},n>\max \{1,2s\},0<s<1,d>0$ and ${\mathcal{N}}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above problem achieves $L^{\mathrm{\infty }}$ bound independent of $d\mathrm{.}$ Using this together with suitable $L^r$-estimates on $u_d,$ we show that least energy solution $u_d$ achieve maximum on the boundary of $\mathrm{\Omega }$ for $d$ sufficiently small.