A nonlinear elliptic problem with terms concentrating in the boundary

Abstract

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 a $ε$-neighborhood of a portion $Γ$ of the boundary. We assume that this $ε$-neighborhood shrinks to $Γ$ as the small parameter $ε$ goes to zero. Also, we suppose the upper boundary of this $ε$-strip presents a highly oscillatory behavior. Our main goal here is 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.