I'm studying a paper and in the introduction appears the following:
It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and extensive literature about critical points which are minimizers, specially for functionals defined on the Sobolev space $W\_{0}^{1,p}(\Omega),\; p>1,$ by
$$J(u)=\int\_{\Omega}\mathcal{F}(x,u,Du) dx,$$
where $\Omega$ is bounded, open subset of $\mathbb{R}^N.$


**DOUBT:** However, I'm struggling to find this extensive literature, and I also would like to find definition and properties of Euler-Lagrange equations.


Thanks in advance.
I appreciate any help.



Definition of Euler-Lagrange equation and properties, where can I find?
These [lecture notes](https://sites.pitt.edu/~hajlasz/Notatki/hel-97.pdf) by Piotr Hajłasz might have the introductory level you are looking for:



> 
> The lectures will be divided into two almost independent streams. One
> of them is the theory of Sobolev spaces with numerous aspects which go
> far beyond the calculus of variations. The second stream is just
> calculus of variations. The theory of Sobolev spaces is a basic
> technical tool for the calculus of variations, however it suffices to
> know only basic results for most of the applications in that context.
> 
> 
> 


