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PDE and mathematical physics · Oct 6, 2026

The Lane–Emden and Hénon–Lane–Emden conjectures

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q>0p,q\gt 0 and real A, B, the system −Δu=∣x∣Avp-\Delta u=|x|^A v^p, −Δv=∣x∣Buq-\Delta v=|x|^B u^q has no positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)>n−2(n+A)/(p+1)+(n+B)/(q+1)\gt n-2, with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and A,B>−2A,B\gt -2.

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