Asymptotic Behaviour of the Least Energy Solutions to Fractional Neumann Problems

We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems: $$ {align*} {cases} { d(-Δ)ˢu+ u= uᵖ⁻¹u } & in Ω, \\ {u>0} & in Ω, \\ { Nₛu=0 } & in Rⁿ Ω̄, {cases} {align*} $$ 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 $u_d$ of the above problem achieve an $L∞$ -bound independent of $d.$ Using this together with suitable $Lʳ$ -estimates on $u_d,$ we show that the least energy solution $u_d$ achieves a maximum on the boundary of $Ω $ for d sufficiently small.