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

TL;DR

This paper investigates the asymptotic behavior of the least energy solutions to a fractional nonlocal Neumann problem, showing boundary concentration and uniform bounds as the parameter d approaches zero.

Contribution

It establishes the boundary maximum principle and uniform bounds for solutions of a fractional Neumann problem as the diffusion parameter diminishes.

Findings

Solutions remain bounded independently of d.

Solutions achieve their maximum on the boundary for small d.

Boundary concentration of solutions as d approaches zero.

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(-\Delta)^{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.