Verón's problem for p-Laplace-type equations

Let $(M^n,g)$ be a closed Riemannian manifold with $\operatorname{Ric}_g\geq (n-1)g$ . For $1<p<n$ , $p<q<p^*:=\frac{np}{n-p}$ , and $\lambda>0$ , determine when every positive solution $u$ of $\Delta_pu-\lambda u^{p-1}+u^{q-1}=0$ on $M$ is the constant solution $u\equiv\lambda^{\frac{1}{q-p}}$ . The cited results show that if $1<p<2$ and $0<\lambda<S_{p,q}^{-1}$ , where $S_{p,q}=\frac{q-p}{2}\left(\frac pn\right)^{\frac p2}\left(\frac{(p^*-1)^2(2-p)}{(p_*-1)(q-1)(p^*-q)}\right)^{\frac{2-p}{2}}$ and $p_*:=\frac{(n-1)p}{n-p}$ , the constant solution is unique, whereas if $2<p<n$ and $p<q<p^*$ , every $\lambda>0$ admits a positive nonconstant solution.

References

Primary source

arXiv

Additional references

Liouville theorem for a class of p-Laplace type equations on manifolds — arXiv — Xi-Nan Ma, Wei Wei, Tian Wu, Hua Zhu

nonlinear-analysis partial-differential-equations

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An August 2026 paper settles the stated question in the advertised parameter ranges: uniqueness holds below the linear case, while above it a positive nonconstant solution exists.

Verón’s uniqueness problem concerns positive solutions of a specified $p$ -Laplace equation on closed manifolds with a Ricci-curvature bound. Ma, Wei, Wu, and Zhu give a parameter-dependent answer in their August 2026 paper.

Known results

$p=2$ , $n\geq 3$ , and $0<\lambda\leq n/(q-2)$ : every positive solution is constant (Bidaut-Véron and Véron).

August 2026 parameter dichotomy

For $1<p<2$ , $p<q<p^*$ , and $0<\lambda<S_{p,q}^{-1}$ , every positive solution is the constant $\lambda^{1/(q-p)}$ . For $2<p<n$ and $p<q<p^*$ , every $\lambda>0$ admits a positive nonconstant solution. Thus the stated problem is resolved in these regimes, with a qualitative transition at $p=2$ .

Current status (as of August 2026): The specified closed-manifold problem is resolved in the paper’s stated parameter ranges; behavior outside those ranges remains unaddressed here.

Sources

arxiv.org

No solutions have been posted yet.