An application of the maximum principle to describe the layer behavior of large solutions and related problems

Abstract

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations Δu = u ^p(x), Δu = −m(x)u + a(x)u ^p(x) where a(x) ≥ a ₀ > 0, p(x) ≥ 1 in Ω, and Δu = e ^p(x) where p(x) ≥ 0 in Ω. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$, while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$. Special emphasis is put in the layer behavior of solutions on the interphase Γ _i : = ∂Ω _c ∩Ω. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider −Δu = λ m(x)u−a(x) u ^p(x) in Ω, u = 0 on ∂Ω, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter λ lies in certain intervals: bifurcation from zero and from infinity arises when λ approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.