For every n≥ 2, 0 0, there do not exist strictly positive functions u,v:R^n→(0,∞) satisfying (-Delta)^s u=v^p and (-Delta)^s v=u^q in R^n whenever frac1p+1+frac1q+1>fracn-2sn.

Fractional Lane–Emden conjecture

For every $n\ge 2$ , $0<s<1$ , and $p,q>0$ , there do not exist strictly positive functions $u,v:\mathbb{R}^n\to(0,\infty)$ satisfying $(-\Delta)^s u=v^p$ and $(-\Delta)^s v=u^q$ in $\mathbb{R}^n$ whenever $\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2s}{n}$ .

References

Primary source

arXiv

Additional references

A proof of the fractional Lane--Emden conjecture — arXiv — Linfeng Mei, Juncheng Wei

Posted 9 hours ago · edited 9 hours ago

Progress summary

Refreshed 9 hours ago

Claimed solved

A new unrefereed paper claims to prove the conjecture’s predicted nonexistence result, but the claim has not been independently checked.

The conjecture predicts that no strictly positive entire solution exists for the fractional Lane–Emden system in the region $\frac{1}{p+1}+\frac{1}{q+1}>\frac{n-2s}{n}$ , without radiality or integrability assumptions.

October 2026 claimed proof

Linfeng Mei and Juncheng Wei claim a proof of nonexistence throughout that region, including nonradial and nonintegrable solutions. The result is presented as an unrefereed preprint and has not been independently assessed.

Current status (as of October 2026): The stated nonexistence result is claimed by Mei and Wei but remains unverified; no independently established resolution is recorded.

Sources

No solutions have been posted yet.