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 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.