2 problems

Lane–Emden conjecture. If $(p,q)$ is subcritical, then the system admits no positive solution in $\left(C^2(\mathbb{R}^N)\right)^2$.

Let $(u,v)$ solve the Lane–Emden system in $\mathbb{R}^n$ with $p\geq q$, and let $\beta$ denote the exponent used in the source's integral estimate. Cheng–Huang–Li conjecture. The…