I have the following convex optimization problem: $$\begin{array}{ll} \text{maximize}_{{q_0,q_1}} & \displaystyle\int_{\Omega} q_1^u{q_0}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\Omega} q_0 \mathrm{d}\mu= 1,\quad \displaystyle\int_{\Omega} q_1\mathrm{d}\mu =1 \\ & f_l \leq {q_0} \leq f_u\\ & g_l \leq q_1 \leq g_u\end{array}$$ where $u\in(0,1) $ and $$\int_{\Omega}f_l \mathrm{d}\mu< 1,\quad\int_{\Omega}g_l \mathrm{d}\mu< 1$$

$$\int_{\Omega}f_u \mathrm{d}\mu> 1,\quad\int_{\Omega}g_u \mathrm{d}\mu> 1$$ Here, $q_0,q_1$ are distinct density functions and $f_l,f_u,g_l,g_u$ are some known positive functions on $\Omega$.

Claim: The solution is unique and it is the same for every $u\in(0,1)$, if $f_u=\infty$ and $g_u=\infty$, i.e., if there are only lower bounds, or $f_l=0$ and $g_l=0$, i.e., if there are only upper bounds. Else, the solution is also unique but it is not the same for all $u$.

Question: Are these claims true, especially the last one?

$\mu$ can be the Lebesgue measure, although I believe the same holds for the counting measure and the discrete sets. The set $\Omega$ can be $\mathbb{R}$ or an interval of real numbers.

I am especially interested in the last ''not'' case, and for this case if necessary $g_u$ and $f_u$ can be assumed to be integrable over $\Omega$.

I had previously asked this question at math.stackexchange but with no answers.