Weak unique continuation for the p-Laplacian in higher dimensions

Establish whether the weak unique continuation property holds for p-harmonic functions, and more generally for solutions of -Δ_pu=f(u), in dimensions N≥3, including the case f=0.

Background

The paper proves weak and strong unique continuation for solutions of -Δ_pu=f(u) in planar domains when 1<p<2. The proof relies on expressing the equation as a determined first-order Beltrami-type system for the complex gradient, a specifically two-dimensional mechanism.

The authors explicitly identify the absence of a corresponding result in dimensions N≥3 as unresolved, even for p-harmonic functions (f=0). They note that a possible route may involve solutions invariant under compact groups of isometries whose orbit space is two-dimensional, such as axially symmetric solutions, because the reduced equation then has a planar structure with a first-order term.