= Solution
Define $A(H)=(H-1)/H^3$ for positive thickness. Its derivative is $A'(H)=(3-2H)/H^4$, so its maximum is attained at $H=3/2$ and equals $4/27$. At $\theta=0$, the equation requires $A(H)=\alpha$, proving the necessary condition $\alpha\le4/27$.
It is also sufficient for a continuous finite periodic branch. On $0<H\le3/2$, $A$ increases monotonically from $-\infty$ to $4/27$, so every value $\alpha\cos\theta$ in the required range has exactly one root on this branch. It joins $H=1$ where $\cos\theta=0$. The second large-thickness root on the ascending side does not extend to a finite periodic branch through those points.
Thus the limiting maximum thickness occurs at the ascending point of the critical film:
$$
\boxed{\alpha_c=\frac4{27},\qquad H_{\max}=\frac32,\qquad
h_{\max}=\frac{3Q}{2U}.}
$$
Subcritical films satisfy a strict thickness bound there. Equivalently the largest carried flux and critical absolute thickness are
$$
Q_c=\frac23\sqrt{\frac{\nu U^3}{g}},\qquad h_c=\sqrt{\frac{\nu U}{g}}.
$$
The mechanism is a transport capacity limit: wall transport grows only as $Uh$, while opposing gravitational drainage grows as $h^3$. At the critical point, adding thickness no longer increases the upward flux. Indeed $\partial Q/\partial h=U(3/H-2)$ on the steady branch and vanishes at $H=3/2$. Excess liquid cannot be maintained as this globally wall-entrained film and must redistribute or drain.
The algebra establishes existence and the limiting branch of the steady lubrication model. A full temporal free-surface stability result requires its time-dependent equation and, when relevant, <pressure> and capillary corrections; the sign of $\partial Q/\partial h$ alone is not a proof of growth of perturbations.
\Image[/past-exam-of-the-flows-fluctuations-and-complexity-course-of-the-university-of-oxford/2016-cylinder-film-transport-limit.png]
{title=Turning point of the coating-film flux relation and the continuous thin-film profiles approaching the critical value alpha = 4/27}
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