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The subcritical Lane–Emden and Hénon–Lane–Emden conjectures
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GAME #370
The subcritical Lane–Emden and Hénon–Lane–Emden conjectures
1 level of pure fluids, heat, waves!
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| 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 $A,B\gt -2$. |
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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.
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