Solvability at the critical source threshold in general domains

Determine whether the Dirichlet problem for the infinite series of p-Laplacians with finite saturation flux is solvable at the critical value of the source parameter in a general domain, and whether the necessary isoperimetric flux condition is sufficient there.

Background

When the saturation flux is finite, the paper proves a necessary condition for weak solvability: the source must satisfy an isoperimetric bound involving the perimeter of every relatively compact finite-perimeter set. For constant sources, this yields a Cheeger-type threshold. The authors establish that the threshold is attained in radial domains and in one dimension, where the necessary condition is also sufficient and the minimizer touches the gradient constraint only on a null set.

The unresolved issue is whether this threshold behavior persists for arbitrary domains and source data. A positive answer would upgrade the necessary condition proved in the paper into a complete solvability characterization, paralleling an open problem cited from earlier work on the corresponding zeroth-order model.