A boundary value problem for a second-order nonlinear equation with delta-like potential

Abstract

A Dirichlet nonlinear problem for a second-order equation is considered on an interval. The problem is perturbed by a delta-like potential ε⁻¹Q(ε⁻¹x), where the function Q(ξ) is compactly supported and 0 < ε ≪ 1. A solution of this boundary value problem is constructed with accuracy up to O(ε) with the use of the method of matched asymptotic expansions. The obtained asymptotic approximation is validated by means of the fixed-point theorem. All types of boundary conditions are considered for a linear boundary value problem.