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Fractional semilinear Neumann problems arising from a fractional Keller--Segel model

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $Ω\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilonΔ)^{1/2}u+u=u^{p},&\hbox{in}~Ω,\\ \partial_νu=0,&\hbox{on}~\partialΩ,\\ u>0,&\hbox{in}~Ω, \end{cases}$$ where $\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small $\varepsilon$, which are obtained by minimizing a suitable energy functional. In the case of large $\varepsilon$ we obtain nonexistence of nonconstant solutions. It is also shown that as $\varepsilon\to0$ the solutions $u_\varepsilon$ tend to zero in measure on $Ω$, while they form spikes in $\overlineΩ$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.