Maximize functional with dot product in it

Question

I have to find a function that maximizes some functional. This function can be divided into multiple functionals - as a first step, I wanted to find a function that maximizes only one of these (for which the correct solution can easily be seen even w/o calculus of variaton), to make sure I got the theory of calculus of variation still right.

The functional I am interested in has the form $$ F[h] = \int_{S \times S} L(x, y, h(x), h(y)) dx dy= \int_{S \times S} g(x, y) h(x)^Th(y) dx dy, $$ with $h:S \to S$ and $g: X \times Y \to R$ and $g$ is symmetrical in its arguments, i.e. $g(x, y) = g(y, x)$ and $S \subseteq R^n$ Now I'd like to find the $h$ that maximizes said functional, i.e. the function $h$ that solves $$ \frac{\delta F}{\delta h} = 0. $$

For this, I tried two things:

1. I used the Euler-Lagrange equation $\frac{\partial F}{\partial h} - \frac{d}{dx}\frac{\partial L}{\partial h'}$, and $\frac{\delta F}{\delta h} = g(x, y)(h(x) + h(y))$, but then I got confused because $h$ is evaluated for two different arguments in $L$. How should I continue from here?
2. I tried to calculate the derivate directly by applying the definition of the functional derivative $$ \frac{\delta F}{\delta h} = \lim_{\epsilon \to 0} \frac{F[h + \epsilon \eta] - F[h]}{\epsilon}. $$ For this I got $$ \lim_{\epsilon \to 0} \int_{S \times S} g(x, y)[(h(x) + \epsilon \eta(x))^T(h(y) + \epsilon \eta(y))] \\ = \int_{S \times S} g(x, y)[\eta(x)^Th(y) + \eta(y)^Tg(x)] \\ = \int_{S \times S} g(x, y)\eta(x)^Th(y) + \int_{S \times S} g(y, x)\eta(y)^Th(x) \\ = \int_{S \times S} 2 g(x, y)h(x)^T\eta(x), $$ from which I concluded $\frac{\delta F}{\delta h} = 2 g(x, y)h(x)^T$ as the expression above must hold true for all test functions $\eta$.

Now I am wondering:

1. Does any of the two approaches make sense and give the correct result
2. How to explain the difference in their results. Is the latter just because I haven't used the symmetry property of $g$ yet?
3. Both expressions I ended up with equal $0$ when $h = const = 0$, which is the global minimum of the functional; how can I find the maximum of it (which should be $h = const = max_S$? I assume the problem here is that I haven't yet used that $h$ should also map to $S$; is that correct? If so, how can I do this?