Following is Proposition 3.3.7 in Bersekas' Nonlinear Programming.
Let $x^*$ be the local minimum of the problem: $$\text{Minimize }\; f(x) $$ $$ \text{subject to: }\ h_j(x) = 0, j=1,\dots, m, g_i(x) \le 0, i=1, \dots, r, $$ where $f,h_j,g_i$ are continuously differentiable functions from $\mathbb{R}^n\to \mathbb{R}$. $h_j$ are linear, $g_i$ are concave. Then there exists $\lambda_1^*,\dots,\lambda_m^*$ and $\mu_1^*,\dots, \mu_r^*$ such that (i)$$ \nabla f(x^*) + \sum_{i=1}^r \mu_i \nabla g_i(x^*) + \sum_{j=1}^m \lambda_j \nabla h_j(x^*) = 0, $$ (ii) $\mu_j^*\ge 0$ (iii) In every neighborhood $N$ of $x^*$, there is an $x$ such that $\lambda_i^*h_i(x) > 0$ for all $i$ with $\lambda_i^* \neq 0$ and $\mu_j^*g_j(x) > 0$ for all $j$ with $\mu_j^* \neq 0$.
My question is, is this theorem applicable even in the infinite dimensional case? Specifically, say, is this applicable in the following setting?
Let $\alpha >1$. Minimize $$\int_{-\infty}^{\infty} p(x)^{\alpha}\, dx,$$ subject to $$\int_{-\infty}^{\infty} p(x)\, dx = 1,$$ $$\int_{-\infty}^{\infty} x\, p(x)\, dx = a_1,$$ $$\int_{-\infty}^{\infty} x^2\, p(x)\, dx = a_2,$$ and $$p(x)\ge 0.\, \forall x$$
Note that the optimizing variable is $p(x)$, not $x$. Unlike the first one, this has a continuum of inequality constraints, namely, $p(x)\ge 0, \, \forall x$ and it is no more an optimization problem in $\mathbb{R}^n$.
Is there any theorem which allows us to apply that the same Prop. 3.3.7 even to this setting?
In order to prove theorems like that one needs the so-called constraint qualifications. Existence of a Slater point if the constraints are convex, for example.
In infinite-dimensional problems similar conditions has to be assumed. On top of the conditions one can expect from finite-dimensional world, there is another type of constraint qualifications involved: certain ranges of of operators have to be closed. (This is not problematic for finite-dimensional problems, where every linear subspace is closed)
Some references could be: