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

Abstract

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, $$\{\begin{array}{ll}(-\epsilon \Delta {)}^{1/2}u+u=u^p,& in\Omega ,\\ {\partial }_{\nu }u=0,& on\partial \Omega ,\\ u>0,& in\Omega ,\end{array}$$ where $\epsilon >0$ 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 $\epsilon$, 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 $\Omega$, while they form spikes in $\overline{\Omega }$. The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.

References in corpus (3)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Existence of solutions for a higher order non-local equation appearing in crack dynamics
• Gradient Estimate on the Neumann Semigroup and Applications