Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

Abstract

AbstractThe structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –ϵΔpu = f(u) in Ω, u = 0 on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied as ϵ → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and $ \underline {\lim} _{u \to 0_{+}} $f(u)/up–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ϵ → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ϵ sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)