I have the following convex optimization problem: $$\begin{array}{ll} \text{maximize}_{{f,g}} & \displaystyle\int_{\Omega} g^u{f}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\Omega} f \mathrm{d}\mu= 1,\quad \displaystyle\int_{\Omega} g\mathrm{d}\mu =1 \\ & f_L \leq {f} \leq f_U\\ & g_L \leq g \leq g_U\end{array}$$ 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, $f$ and $g$ 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.

Addendum: For $f_U=\infty$ and $g_U=\infty$ I know that $$\frac{g}{f}(y)=\begin{cases}c_1\quad\mbox{if}\quad g/f<c_1\\ h(y)\quad\mbox{if}\quad c_1\leq g/f \leq c_2\\ c_2\quad\mbox{if}\quad g/f>c_2\\\end{cases}$$ where $c_1$ and $c_2$ are some constants an $h$ is a function of the upper and lower bounding functions for example $(g_U+g_L)/(f_U+f_L)$. I think the same is true for the case $f_L=0$ and $g_L=0$.