Nonexistence conjecture for the Lane–Emden system in strictly starshaped domains

Let $\Omega$ be a bounded domain, strictly starshaped with respect to $0\in\Omega$, and suppose the condition

$$holds for the Lane–Emden system$$

. Consider the problem with $u=v=0$ on $\partial\Omega$. Lane–Emden nonexistence conjecture. The problem does not admit any nontrivial solution. The conjecture concerns the still-open nonexistence of nontrivial solutions for the Lane–Emden system in a strictly starshaped domain, extending the type of conclusion suggested by the preceding unique-continuation and Pohozaev-type considerations.