Phase Transition Solutions With Spike For Fife-Greenlee Problem

We consider t.he Fife-Greenlee problemε2Δu+(u-a(y))(1-u2)=0 in Ω,(?)=0 on(?)Ω,where Ω is a bounded domain in R2 with smooth boundary,ε>0 is a small parameter,n denotes the unit outward normal of(?)Ω.Assume that Γ = { y∈Ω:a(y)= 0} is a simple smooth curve intersecting orthogonally with(?)Ω at exactly two points,saying P1,P2,and dividing Q into two disjoint,nonempty components,saying Ω+ and Ω_.We assume that-1<a(y)<1 on Ω and ▽a≠0 on Γ,and also some admissibility conditions between the curves Γ,(?)Ω and the inhomogeneity a(y)hold at P1,P2.We will show the existence of a solution uε with a transition layer near Γ and a downward spike near the maximum points of a(y)whose profile looks like uε→C<1 at a point Pε,uε→+1 in Ω+\Pε,uε→-1 in Ω_,as ε→0.