A class of boundary traces for solutions of the equation \(Lu=\psi (u)\) (Q2461254)

From the authors' abstract: Our subject is the class \({\mathcal U}\) of all positive solutions of a semilinear equation \(Lu=\psi(u)\) in \(E\) where \(L\) is a second order elliptic differential operator, \(E\) is a domain in \(\mathbb{R}^d\) and \(\psi\) belongs to a convex class \(\Psi\) of \(C^1\) functions which contains functions \(\psi(u)=u^\alpha\) with \(\alpha>1\). A special role is played by a class \({\mathcal U}_0\) of solutions which we call \(\sigma\)-moderate. A solution \(u\) is moderate if there exists \(h\geq u\) such that \(Lh=0\) in \(E\). We say that \(u\in{\mathcal U}\) is \(\sigma\)-moderate if \(u\) is the limit of an increasing sequence of moderate solutions. In the present paper we develop a general scheme to deal the fine and precise traces. An implication of this equivalence is a Wiener type criterion for vanishing of the Poisson kernel of the equation \(Lu(x)=a(x)u(x)\).