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

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, $$\{\begin{array}{ll}(-\epsilon \mathrm{\Delta }{)}^{1\mathrm{/}2}u+u=u^p,& \text{in}\mathrm{\Omega },\\ {\mathrm{\partial }}_{\nu }u=0,& \text{on}\mathrm{\partial }\mathrm{\Omega },\\ u>0,& \text{in}\mathrm{\Omega },\end{array}$$ where $\epsilon >0$ and $1<p<(n+1)\mathrm{/}(n-1)$. 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 $\epsilon$, which are obtained by minimizing a suitable energy functional. In the case of large $\epsilon$ we obtain nonexistence of nonconstant solutions. It is also shown that as $\epsilon \to 0$ the solutions $u_{\epsilon }$ tend to zero in measure on $\mathrm{\Omega }$, while they form spikes in $\bar{\mathrm{\Omega }}$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.