Nonexistence of positive very weak solutions to an elliptic problem with boundary reactions (Q487291)

The paper studies positive solutions of the nonlinear Robin problem for the Laplace equation \(\Delta u=0\) in \(\Omega \), \(\partial u/\partial \nu +u=au^p+f\) on \(\partial \Omega \). Here \(\Omega \subset \mathbb{R}^N\), \(N\geq 3\), is a bounded domain with smooth boundary, \(p>1\), \(f\in L^\infty (\partial \Omega )\), \(f\geq 0\), \(f\not \equiv 0\), \(a\in L^1(\partial \Omega )\), \(a\geq 0\). Under assumption that \(0\in \partial \Omega \) and \(\int_{\partial \Omega }a(y) |y|^{2-N}\;ds(y)=\infty \) it is shown that there is no positive solution of the problem even in the very weak sense.