Boundary layer and spike layer solutions for a bistable elliptic problem with generalized boundary conditions

We show that for large λ>0λ>0, the “generalized” boundary value problem-Δu=λu(u-a(x))(1-u)in Ω,u|∂Ω=φ,where 1⩽φ(x)⩽∞1⩽φ(x)⩽∞, behaves like the special case φ≡1φ≡1. In particular, we show the existence of solutions with sharp boundary layers and interior spikes, and determine the location of the spikes.