The Lane–Emden and Hénon–Lane–Emden conjectures

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, $p,q\gt 0$ and real A, B, the system $-\Delta u=|x|^A v^p$, $-\Delta v=|x|^B u^q$ has no positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)\gt n-2$, 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 $A,B\gt -2$.