Monotonicity and nonexistence conjecture for the Emden–Fowler equation

Let $N\ge 3$, let $p>p_{\rm S}$, and let $\Theta(\Gamma)$ denote the relevant solution-branch parameter for the Emden–Fowler equation

. Let $p_{\rm JL}$ and $\Theta^*$ be the Joseph–Lundgren exponent and the angle given in Theorem C, respectively. **Monotonicity and nonexistence conjecture.** If $N\ge 11$ and $p\ge p_{\rm JL}$, then $\Theta(\Gamma)$ is strictly decreasing and

has no regular solution for $\Theta\in(0,\Theta^*]$. This would clarify the behavior of the solution branch in the case $p\ge p_{\rm JL}$, which the surrounding theorems leave unresolved.