On rapidly oscillating solutions of a nonlinear elliptic equation

Abstract

The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation ε²△u = f(u). We describe the asymptotic behavior as ε tends to zero of the solutions on a spherical crown C of R^N, (N ≥ 2) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics.Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.