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Fractional Lane-Emden Conjecture

The fractional Lane-Emden conjecture is a nonexistence assertion for positive entire solutions of the system

(-Delta)^su=v^p
(1)
(-Delta)^sv=u^q,
(2)

where n>=2, 0<s<1, p,q>0, and (-Delta)^s is the fractional Laplacian. Its subcritical condition is 1/(p+1)+1/(q+1)>(n-2s)/n.

Mei and Wei (2026) reported that under this condition there are no finite continuous positive functions on R^n satisfying the pointwise integral system

u(x)=c_(n,s)int_(R^n)(v(y)^p)/(||x-y||^(n-2s))dy
(3)
v(x)=c_(n,s)int_(R^n)(u(y)^q)/(||x-y||^(n-2s))dy,
(4)

where c_(n,s) is the positive normalization constant of the Riesz potential of order 2s. The corresponding differential system has no positive solution with u,v in C_(loc)^(1,1)(R^n) and int_(R^n)(|u(x)|+|v(x)|)/(1+||x||^(n+2s))dx<infty.

This is an entire-space fractional system, distinct from the radial Lane-Emden differential equation.

See also

Fractional Laplacian, Lane-Emden Differential Equation, Riesz Potential

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References

Mei, L. and Wei, J. "A Proof of the Fractional Lane-Emden Conjecture." 7 Oct 2026. https://arxiv.org/abs/2610.10188.

Cite this as:

Weisstein, Eric W. "Fractional Lane-Emden Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FractionalLane-EmdenConjecture.html

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