OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q> 0 and real A, B, the system -\Delta u=|x|^A v^p, -\Delta v=|x|^B u^q has no positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)> 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> -2.
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MANUSCRIPT 01
The Subcritical Hénon–Lane–Emden Conjecture
We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)> n-2. Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3…
Full source abstract
We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)> n-2. Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and A,B> -2, combining this result with the radial existence theorem of Bidaut-Véron and Giacomini proves Phan's Conjecture C in full: a positive radial entire solution exists whenever the strict inequality fails.
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