Family 370Partial differential equationsLeanNot yet digested
The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, and real A, B, the system , has no positive entire solution when , 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 .
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The Subcritical Hénon–Lane–Emden Conjecture
Sep 24, 2026·paper.pdfLean
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 . Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and , 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.