Lane–Emden conjecture

The Lane–Emden conjecture concerns nonexistence of positive entire solutions to the coupled elliptic system in $\mathbb{R}^n$. The complete conjecture is known in dimensions $3$ and $4$, but remains unresolved in higher dimensions.

Known results

• Mitidieri settled the radial case.
• Serrin and Zou proved conditional results under polynomial-growth assumptions.
• Poláčik, Quittner, and Souplet proved the dimension-$3$ case without growth assumptions.
• Souplet completed the dimension-$4$ case; the result was published on March 19, 2009, in Advances in Mathematics.

October 2026 new Liouville region

Haoyang Lu and Zhitao Zhang report nonexistence, without growth or decay assumptions, when $n\ge 5$, $p\ge 1$, $q\ge 1$, and $\frac{1}{p+1}+\frac{1}{q+1}\ge 1-\frac{2}{n}+\frac{4}{n^2}$. They describe this as a partial enlargement of the known region, not a proof of the critical-hyperbola conjecture.

Current status (as of October 2026): The conjecture is settled for $n=3$ and $n=4$ and partially resolved for $n\ge 5$; the full higher-dimensional conjecture remains open.