Monotone discontinuity of lattice operations in a quasilinear harmonic space (Q1816456)

The solutions of a second order quasilinear elliptic equation \(-\text{div} {\mathcal A}(x,\nabla u(x))=0\) are called \({\mathcal A}\)-harmonic because they have many features in common with harmonic functions [see \textit{J. Heinonen, T. Kilpeläinen} and \textit{O. Martio}, Nonlinear potential theory of degenerate elliptic equations (Oxford Univ. Press) (1993; Zbl 0780.31001)]. The order structure and the induced lattice structure of the space of \({\mathcal A}\)-harmonic functions supplement the lack of a linear structure. An example is given by the weak solutions of the \(p\)-Laplace equation \(-\text{div} (|\nabla u|^{p-2}\nabla u)=0\). In the above reference the properties of the set of \({\mathcal A}\)-harmonic functions have been axiomatized. An open problem was: Let \((u_n)_{n\in\mathbb{N}}\) be an increasing sequence of \({\mathcal A}\)-harmonic functions on a region \(\Omega\subset\mathbb{R}^m\). Denote the greatest common \({\mathcal A}\)-harmonic minorant of \(u_n\) and \(0\) by \(u\wedge 0\) for all \(n\in\mathbb{N}\). Let \((u_n)\) converge to \(u_\infty\) in \(\Omega\). Then \((u_n\wedge 0)_{n\in\mathbb{N}}\) converges to \(u_\infty\wedge 0\) in \(\Omega\). The author gives a counterexample to this continuity statement. Hence the set of axioms in the above reference does not allow to verify it.