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 in Ω, $\frac{\partial u}{\partial \nu}$ + u = a(x) u^p + f(x) on ∂Ω,
where Ω ⊂ R^N, N ≥ 3, is a smooth bounded domain with a flat boundary portion, p > 1, a, f ∈ 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 ∈ L^∞(∂Ω), f $\not\equiv$ 0, even in the very weak sense.