Solutions with spikes for quasilinear boundary value problems

Abstract

In recent years there has been some progress in the study of spike layer solutions for semilinear singular perturbation problems. Spike layer behavior can be described as nonuniform limiting behavior of a solution in which the solution has an interior maximum or minimum inside the layer. One of the first rigorous studies was made by O’Malley [lo], who used phase plane analysis for the autonomous semilinear Dirichlet problem and showed that solutions exist with increasing numbers of spikes as the small parameter approaches zero and that the spikes occur at equally spaced points in the interval. More recently, various results for nonautonomous ordinary differential equations with Dirichlet or Neumann boundary conditions have been obtained by Kath [4], Lange [7], Butuzov and Vasil’eva [ 11, and Kurland [6]. Spike layer solutions for semilinear elliptic partial differential equations have been treated by Lin, Ni, and Takagi [8] for Neumann boundary conditions and by Kelley and Ko [S] for Dirichlet boundary conditions. We consider in this paper the quasilinear problem