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We consider the following fractional semilinear Neumann problem on a smoothbounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases}(-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nuu=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ where$\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of thesemilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. Theproblem arises by considering steady states of the Keller--Segel model withnonlocal chemical concentration diffusion. Using the semigroup language for theextension method and variational techniques, we prove existence of nonconstantsmooth solutions for small $\varepsilon$, which are obtained by minimizing asuitable energy functional. In the case of large $\varepsilon$ we obtainnonexistence of nonconstant solutions. It is also shown that as$\varepsilon\to0$ the solutions $u_\varepsilon$ tend to zero in measure on$\Omega$, while they form spikes in $\overline{\Omega}$. The regularityestimates of the fractional Neumann Laplacian that we develop here areessential for the analysis. The latter results are of independent interest.

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