Prof. Zhitao Zhang and his former student Dr. Kui Li have recently published one paper entitled “Proof of the Hénon-Lane-Emden conjecture in R3” in J. Differential Equations (https://doi.org/10.1016/j.jde.2018.07.036).
The Hénon-Lane-Emden conjecture states that there is no non-trivial non-negative solution for the Hénon-Lane-Emden elliptic system whenever the pair(p,q) of exponents is subcritical
This is an important conjecture in the field of nonlinear partial differential equations, which has been a challenging and difficult problem for many years because one of the exponents may be supcritical. Bidaut-Veron and Giacomini, Adv. Differential Equations 15, (2010) showed that the system had positive radial solutions if and only if the pair (p, q) is above or on the critical Sobolev hyperbola, which implies this conjecture is true for radial positive solutions. Under the assumption on the bounds of positive solutions, Fazly and Ghoussoub (Fellow of the Royal Society of Canada, Fellow of the American Mathematical Society), Discrete Contin. Dyn. Syst. 34 (2014) proved this conjecture for R3.
Prof. Zhang and Dr. Li have proved completely this conjecture is true for space dimension N=3 by scale invariance of the solutions and Sobolev embedding on SN-1 , which also implies the single elliptic equation has no positive classical solutions in R3 when the exponent lies below the Hardy-Sobolev exponent, this also covers the conjecture of Phan-Souplet, J. Differential Equations 252, (2012) for R3.
The referee said:“This is an important conjecture in the theory of PDEs, and in the last several decades, many mathematicians tried to solve it with little progress... This result is clean, neat, and extremely interesting, and the article is very well written. Hence I warmly recommend the article for publication in JDE”.
Academy of Mathematics and Systems Science, CAS