Proof of the Hénon–Lane–Emden conjecture in [formula omitted

Abstract

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on SN−1, we prove this conjecture is true for space dimension N=3; which also implies the single elliptic equation has no positive classical solutions in R3 when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for R3.