Multiplicity and Attainment for Critical Hardy p-Laplacian Problems

Definitions and mathematical statement

For $1<p<\infty$, the $p$-Laplacian is \[ \Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u). \] The verified source question has two parts: describe the multiplicity of solutions of its $p$-Laplacian equation with critical Hardy potential, including the case where $\Omega$ is a ball; and prove or disprove the converse of its stated sufficient criterion for attainment of $\lambda_p(\Omega)$.

A critical Hardy potential refers to a singular potential coupled at the threshold in an appropriate Hardy inequality. For a specified weight $V_H$ and energy class, such a threshold is described by the best constant $C_H$ in \[ \int_\Omega|\nabla u|^p\,{\,\mathrm{d}} x \ge C_H\int_\Omega V_H|u|^p\,{\,\mathrm{d}} x. \] This explains the terminology only: the extract does not identify the weight, singularity location, critical constant, full equation, nonlinear right-hand side, boundary data or solution class. Interior singularities and boundary-distance Hardy potentials are different problems, so none is silently chosen.

Likewise, the variational formula defining $\lambda_p(\Omega)$ and the sufficient attainment criterion are absent. Attainment means that a nonzero admissible function realizes the specified infimum, but the requested converse cannot be written as a definite equivalence until both that infimum and its sufficient hypothesis are recovered. In particular, $\lambda_p(\Omega)$ is not assumed to be the first ordinary Dirichlet eigenvalue or the Hardy constant $C_H$.

The literature assessment checked on 21 August 2026 addresses the converse-attainment part and says no authoritative proof or counterexample to that exact converse was found. It supplies no multiplicity classification, even for balls, and no full theorem to reverse. The open label is dated triage, not independent verification of a uniquely specified general PDE theorem.

Short English statement

Determine multiplicity for the report's critical-Hardy $p$-Laplacian problem, including ball domains, and test the converse of its attainment criterion for $\lambda_p(\Omega)$. The precise equation, variational constant and criterion are missing and must be recovered before either part becomes a full theorem.

Sources

• Topological and Variational Methods for Differential Equations (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/29, pp. 1601--1678, https://doi.org/10.4171/owr/2005/29; proposed year supplied as 2005.
• The sole literature source in the assessment checked 21 August 2026 is the "original problem report," https://doi.org/10.4171/owr/2005/29. It supplies no later proof or counterexample for the exact converse.
• Local UnsolvedMath registry, permanent ID $30000260$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The clean statement is labelled corrected and verified through restoration of balanced mathematical notation, retaining both source questions.