TY - JOUR
AU - García-Melián, J.
AU - Rossi, J.D.
AU - Sabina de Lis, J.C.
T1 - An application of the maximum principle to describe the layer behavior of large solutions and related problems
LA - eng
PY - 2011
SP - 183
EP - 214
T2 - Manuscripta Mathematica
SN - 0025-2611
VL - 134
IS - 1
AB - 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 = up(x), Δu = -m(x)u + a(x)up(x) where a(x) ≥ a0 > 0, p(x) ≥ 1 in Ω, and Δu = ep(x) where p(x) ≥ 0 in Ω. In the first two cases p is allowed to take the value 1 in a whole subdomain Ωc ⊂ Ω, while in the last case p can vanish in a whole subdomain Ωc ⊂ Ω. Special emphasis is put in the layer behavior of solutions on the interphase Γi: = ∂Ωc∩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) up(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. © 2010 Springer-Verlag.
DO - 10.1007/S00229-010-0391-Z
UR - https://portalciencia.ull.es/documentos/5e3c36d029995246bbf5e247
DP - Dialnet - Portal de la Investigación
ER -