Lane–Emden conjecture for the subcritical Lane–Emden system

Let $N\in\mathbb{N}_+$ and $p,q>0$. Consider the Lane–Emden system

$$\left\{ \begin{aligned} -\Delta u&=|v|^{p-1}v \\ -\Delta v&=|u|^{q-1}u \end{aligned} \quad\text{in }\mathbb{R}^N. \right.$$

A pair $(p,q)$ is subcritical when

$$\frac{N}{p+1}+\frac{N}{q+1}>N-2.$$

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

P. Souplet completely resolved the conjecture for $N\leq 4$ and obtained substantial partial results for $N\geq 5$. It remains open in higher dimensions.