Let us consider a Hilbert space $H$ (with infinite dimension) and a functional $J : H \longrightarrow \mathbb R_+$.
Do we have a characterization of the minimums of $J$ with the differential of $J$ (Fréchet or Gateaux differential for example) ? I do not have any simple reference when the space has an infinite dimension. Do we have a theorem of Lagrange multipliers for the differential with a constraint ?
This question is linked with the following problem. Let us consider $H = H^1(\mathbb R)$ and $J : u \mapsto \| u\|_H$. We want to minimize $J$ over the constraint $\int_{\mathbb{R}} u^n = \| u\|^2_H$ where $n$ is a fixed positive integer.
Another question : does the minimum of $J$ remains the same if we relax the condition $\int_{\mathbb{R}} u^n \geqslant \| u\|_H$ (a sort of maximum principle for that type of functional) ?