On a singularly perturbed semi-linear problem with robin boundary conditions

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition: ^(ε ∇ ^∆ w ^w · ⁻ ñ + ^λw γw ^1+χ = ⁼ 0, ⁰ on ^{in Ω} ∂Ω (∗) where Ω ⊂ R^N(N ≥ 1) is a bounded domain with smooth boundary, ñ denotes the unit outward normal vector of ∂Ω and γ ∈ R/{0}. ε and λ are positive constants. The problem (∗) is derived from the well-known singular Keller-Segel system. When γ > 0, we show there is only trivial solution w = 0. When γ < 0 and Ω = B_R(0) is a ball, we show that problem (∗) has a non-constant solution which converges to zero uniformly as ε tends to zero. The main idea of this paper is to transform the Robin problem (∗) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (∗), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system.