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: {array(−ϵΔ)1/2u+u=h(u)arrayinΩ,array∂νu=0arrayon∂Ω,$$ \left\{\begin{array}{cc}{\left(-\epsilon \Delta \right)}^{1/2}u+u=h(u)\kern0.60em & \kern0.1em \mathrm{in}\kern0.5em \Omega, \\ {}{\partial}_{\nu u}=0\kern0.30em & \mathrm{on}\kern0.5em \mathrm{\partial \Omega },\end{array}\right. $$ where normalΩ⊂ℝn0.1emfalse(n≥2false)$$ \Omega \subset {\mathbb{R}}^n\kern0.1em \left(n\ge 2\right) $$ is a smooth bounded domain and ν$$ \nu $$ is the outward unit normal to ∂normalΩ$$ \mathrm{\partial \Omega } $$. First, under the superlinear and subcritical growth assumptions on h$$ h $$, we prove that there exists at least one positive nonconstant solution uϵ$$ {u}_{\epsilon } $$ for small ϵ>0$$ \epsilon >0 $$ and the family of solutions false{uϵfalse}ϵ>0$$ {\left\{{u}_{\epsilon}\right\}}_{\epsilon >0} $$ is uniformly bounded. Moreover, we build a Pohozaev‐type identity for the ( ϵ$$ \epsilon $$‐) Neumann harmonic extension of the following problem: {array(−ϵΔ)1/2u=g(u)arrayinΩ,array∂νu=0arrayon∂Ω,$$ \left\{\begin{array}{cc}{\left(-\epsilon \Delta \right)}^{1/2}u=g(u)\kern0.60em & \kern0.1em \mathrm{in}\kern0.5em \Omega, \\ {}{\partial}_{\nu u}=0\kern0.30em & \mathrm{on}\kern0.5em \mathrm{\partial \Omega },\end{array}\right. $$ where g$$ g $$ is a C1$$ {C}^1 $$ function such that gfalse(0false)=0$$ g(0)=0 $$. As a direct application of this identity, when g$$ g $$ satisfies false(n−1false)tgfalse(tfalse)−2nGfalse(tfalse)≥0$$ \left(n-1\right) tg(t)-2 nG(t)\ge 0 $$, where Gfalse(tfalse)=∫0tgfalse(sfalse)ds$$ G(t)={\int}_0^tg(s) ds $$, we deduce the nonexistence of weak bounded solution in star‐shaped domains.