Remark: Family 370. The subcritical Lane–Emden and Hénon–Lane–Emden conjectures

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For $n\ge 2$, $p,q>0$ 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 $n\ge 3$ and $A,B>-2$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.