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 S N − 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 R 3 when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for R 3. [ABSTRACT FROM AUTHOR]