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: \begin{align*} \left\{\begin{array}{l l} { d(-Δ)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $\Omega ,$ } { \mathcal{N}_{s}u=0 } \text{ in ${\mathbb{R}}^n\setminus \overline{\Omega },$} {u>0} \text{ in $\Omega ,$} \end{array} \right.\end{align*} where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain of class $C^{1,1}$, $1<p<\frac{n+s}{n-s},n>\max \left\{1,2s\right\},0<s<1,d>0$ 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 $L^{\infty }$ bound independent of $d.$ 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 $\Omega$ for $d$ sufficiently small.

23 pages