I unsuccessfully looked for a similar topic to the one I am submitting to your attention. My question is the following.
Let $f:[0,\infty[^d \to \mathbb R$ and $g : [0, \infty[^d \to \mathbb{R}$ with $f,g \in \mathcal C^1$. Both $f$ and $g$ are strictly concave and admit a unique global maximum. Denote $h(x) \equiv f(x) + g(x)$. I know from other topics that it can be shown that $\max h(x) \leq \max f(x) + \max g(x)$. Can I conclude something about the existence of a maximum of $h(x)$? I know that this maximum does not need to exist in general but, given my assumptions, is there any condition I can apply to show existence? Unfortunately, I cannot use Weierstrass since my constraint set is not compact. In case, could you provide me with some reference I can look at?
Thanks to anyone who will try to address this question!