I've been studying Variational Calculs from Differential Equations and Variational Calculus by L. Elsgoltz. Everything is ok, but the whole theory is developed bounded intervals of the real numbers, so I'm asking for a reference book where I can study the theory on unbounded intervals or even unbounded domains of $\mathbb{R}^n$. Can one adaptated the same proofs of the bounded case, but now using Dominated convergence theorem to interchange the sign integral and the derivative? It would require stronger hypothesis, but I think it would work.
Thanks in advance!