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 $(n+A)/(p+1)+(n+B)/(q+1)\gt n-2$. Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and $A,B\gt -2$, 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.
Comments and reviews
No comments yet.
API
curl /api/v1/records/openai-math/The-Subcritical-Henon-Lane-Emden-Conjecture
curl -X POST /api/v1/records/openai-math/The-Subcritical-Henon-Lane-Emden-Conjecture/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/openai-math/The-Subcritical-Henon-Lane-Emden-Conjecture/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'