Nonlinear problems with solutions exhibiting a free boundary on the boundary

Abstract

We prove existence of nonnegative solutions to -Delta u + u = 0 on a smooth bounded domain Omega subject to the singular boundary derivative condition partial derivative u/partial derivative v = -u(-beta) + lambda f (x, u) on partial derivative Omega boolean AND {u > 0} with 0 < beta < 1. There is a constant lambda* such that for 0 < lambda < lambda* every nonnegative solution vanishes on a subset of the boundary with positive surface measure. For lambda > lambda* we show the existence of a maximal positive solution. We analyze its linearized stability and its regularity. Minimizers of the energy functional related to the problem are shown to be regular and satisfy the equation together with the boundary condition.