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

Abstract

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