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

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form – ϵ Δ p u = f ( u ) in Ω, u = 0 on ∂Ω, Ω ⊂ R N a bounded smooth domain, is studied as ϵ → 0 + , for a class of nonlinearities f ( u ) satisfying f (0) = f ( z 1 ) = f ( z 2 ) = 0 with 0 < z 1 < z 2 , f < 0 in (0, z 1 ), f > 0 in ( z 1 , z 2 ) and $ \underline {\lim} _{u \to 0_{+}} $ f ( u )/ u p –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)