[A duplicate thread can also be found at https://stats.stackexchange.com/questions/59450/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ]

Let $F(\alpha) := \mathbb{P}(\tilde{\alpha} \leq \alpha)$ be an arbitrary, strictly increasing and twice differentiable CDF that is defined on the interval $[0, \overline{\alpha}]$, where $\overline{\alpha}>1$ may be infinite. Moreover, let $\mathbb{E}(\tilde{\alpha}) = 1$.

Let $N \geq 2$ be a natural number, and $\delta \in (0,1)$ real.

I am looking for necessary and sufficient criteria on $F$ in order to find a solution to

$(N-1)[1-\delta+\delta F(\alpha)] - \delta \alpha F'(\alpha) =0$,

where $\alpha$ can be any real positive number not larger than $\overline{\alpha}$.

Now, it is clear that the left term, $(N-1)[1-\delta+\delta F(\alpha)]$, is strictly increasing in $\alpha$ and bounded above by $N-1$. Hence, if the probability mass is sufficiently "concentrated" in some interval (implying that $F'(.)$ is large in that interval), a solution $\alpha^*$ must exist by continuity of $F'$ and $F$.

However, it would be nice to have some sharper conditions on $F$ that are necessary or sufficient for a solution. Ideally, I'd wish to have a result that gives an upper bound on $F$'s variance, or similar.

One thing that seems problematic is that the above equation never uses the fact that $\mathbb{E}(\tilde{\alpha})=1$. I've tried to apply Markov's inequality but it didn't help me much.

Numerical simulations reveal that a solution can usually be found if the variance of the considered CDF is sufficiently low. Examples include $F$ log-normal or $F(\alpha) = (\frac{b \alpha}{b+1})^b$ for $b>1$. The uniform distribution on $[0,2]$ is a borderline case that doesn't permit a solution.

I would greatly appreciate any help or ideas to this (economics) research problem. Many thanks!