Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions

Abstract

We consider a semilinear elliptic problem with the boundary reaction: $$−Δu = 0 \quad\mathrm{in}\quad Ω, \quad\frac{\partial u}{\partial \nu} + u = a(x) u^p + f(x) \quad\mathrm{on}\quad ∂Ω,$$ where Ω $\subset$ R^N, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f $\in$ L¹(∂Ω) are nonnegative functions, not identically equal to zero. We provide a necessary condition and a sufficient condition for the existence of positive very weak solutions of the problem. As a corollary, under some assumption of the potential function a, we prove that the problem has no positive solution for any nonnegative external force f $\in$ L^∞(∂Ω), f $\not\equiv$ 0, even in the very weak sense.