On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

Abstract

In this paper, we study the following fractional semilinear Neumann problem arising from Keller–Segel model: where is a smooth bounded domain and is the outward unit normal to . First, under the superlinear and subcritical growth assumptions on , we prove that there exists at least one positive nonconstant solution for small and the family of solutions is uniformly bounded. Moreover, we build a Pohozaev‐type identity for the ( ‐) Neumann harmonic extension of the following problem: where is a function such that . As a direct application of this identity, when satisfies , where , we deduce the nonexistence of weak bounded solution in star‐shaped domains.