Let M be a complete Riemannian manifold satisfying the geometric hypotheses in Holopainen’s conjecture. Is every nonnegative p-harmonic function v on M such that v∈ L^p-1(M) constant; equivalently, does Delta_p v=0 and int_M v^p-1,d mu<∞ imply that v is constant?

Holopainen’s L^p−1-Liouville conjecture

Holopainen’s conjecture concerns Liouville properties for nonlinear diffusion under geometric growth conditions. The latest preprint explicitly presents its Liouville theorem as only a partial answer, not a resolution.

Known results

No additional classical partial results were identified in the retrieved sources.

September 2026 partial result

The preprint claims mass conservation in two parameter regimes, finite-time extinction on hyperbolic space in the complementary regime, and an $L^{p-1}$ Liouville property at the critical exponent. These results advance the conjecture but do not settle it; the claims remain unverified here.

Current status (as of September 2026): The preprint claims a partial result, while the full Holopainen conjecture remains open and unverified.

Holopainen’s L^{p−1}-Liouville conjecture

Primary source

Additional references

Progress summary

Solutions 0