A nonlinear elliptic problem with terms concentrating in the boundary
Gleiciane S. Aragão, Antônio Luíz Pereira, Marcone Corrêa Pereira · Mathematical Methods in the Applied Sciences · 2012
In this paper, we investigate the behavior of a family of steady‐state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in aε‐neighborhood of a portion Γ of the boundary. We assume that thisε‐neighborhood shrinks to Γ as the small parameterεgoes to zero. Also, we suppose the upper boundary of thisε‐strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.