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

In this paper, we study the following fractional semilinear Neumann problem arising from Keller-Segel model: {(-epsilon Delta)(1/2)u + u = h(u) in Omega, partial derivative(nu u) = 0 on partial derivative Omega, where Omega subset of R-n (n >= 2) is a smooth bounded domain and nu is the outward unit normal to partial derivative Omega. First, under the superlinear and subcritical growth assumptions on h, we prove that there exists at least one positive nonconstant solution u(epsilon). for small epsilon > 0 and the family of solutions {u(epsilon)}(epsilon>0) is uniformly bounded. Moreover, we build a Pohozaev-type identity for the (epsilon-) Neumann harmonic extension of the following problem {(-epsilon Delta)(1/2)u = g(u) in Omega, partial derivative(nu u) = 0 on partial derivative Omega, where g is a C-1 function such that g(0) = 0. As a direct application of this identity, when g satisfies (n - 1)tg(t) - 2nG(t) >= 0, where G(t) = integral(t)(0)g(s)ds, we deduce the nonexistence of weak bounded solution in star-shaped domains.