Solutions with transition layer and spike in an inhomogeneous phase transition model

We consider the following singularly perturbed elliptic problemε2Δu˜+(u˜−a(y˜))(1−u˜2)=0inΩ,∂u˜∂n=0on∂Ω, where Ω   is a bounded domain in R2R2 with smooth boundary, −10} and Ω−={y˜∈Ω:a(y˜)<0} and ∂a∂ν>0 on Γ, where ν   is the outer normal to Ω−Ω−. We will show the existence of a solution uεuε with a transition layer near Γ   and a downward spike near the maximum points of a(y˜) whose profile looks likeuε→C<1at a pointPε,uε→1inΩ+∖Pε,uε→−1inΩ−,asε→0.