The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems

Abstract:

We consider the nonlinear singularly perturbed problem \[ -\epsilon ^2\Delta u = f(u), \quad u > 0 \quad \mbox {in} \quad \Omega , u = 0 \quad \mbox {on} \quad \partial \Omega , \] where $\Omega \subset {\mathbf {R}}^N$ ($N \ge 2$) is an appropriately smooth bounded domain and $\epsilon > 0$ is a small parameter. It is known that under some conditions on $f$, the solution $u_\epsilon$ corresponding to $\epsilon$ develops boundary layers when $\epsilon \to 0$. We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.