Asymptotic Spatial Patterns and Entire Solutions of Semilinear Elliptic Equations

(2) ε∆uε + f(uε) = 0, x ∈ Ω, Bu = 0, x ∈ ∂Ω, where ε > 0 is a small parameter, Ω is a smooth bounded domain in Rn, and Bu is an appropriate boundary condition. The connection of (1) and (2) are made by a typical technique called blowup method. Suppose that {uε} is a family of solutions of (2). The simplest setup of the blowup method is to choose Pε ∈ Ω, and define vε(y) = uε(εy + Pε), for y ∈ Ωε = {y : εy + Pε ∈ Ω}. Then usually in a proper sense, vε (y) → U(y), as ε → 0, where U is an entire solution if Pε is not too close to ∂Ω. (See for example, Gidas and Spruck [GS2], and Ni and Takagi [NT1, NT2].) Thus the local spatial pattern of the equilibrium solution to the reaction-diffusion equation is governed by the spatial pattern of the entire solution. On the other hand, if uε is bounded and nonnegative, then it is also natural to require the entire solution to be bounded and nonnegative.