The Subcritical Hénon--Lane--Emden Conjecture

We prove the subcritical Hénon--Lane--Emden conjecture: for every dimension $n\ge2$, 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 \(n\ge3\) 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.