# Definition of Euler-Lagrange equation and properties, where can I find?

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.

• I think any calculus of variations book will have this result, try Dacorogna's Introduction to Calculus of Variations for example. Nov 25 at 1:01
• @CheeHan I've studied with this exact book. However, in that book, they assume we have the background I'm asking here. Nov 25 at 1:16
• By background, you meant Sobolev spaces? Nov 25 at 1:18
• If that's the case, then try Dacorogna's Direct Method in the Calculus of Variations. Nov 25 at 1:20
• No, I've read that too. By background, I mean how to build an Euler-Lagrange Equation from a given Functional in Sobolev Spaces. Nov 25 at 18:02

• Yes, this will help. I start reading, these lecture notes might help with another good book I found, which is: Semilinear Elliptic Equations for Beginners (Marino Badiale and Enrico Serra). They explain even how $-div$ appears at the beginning of the Euler-Lagrange equations. Thank you very much for your help. Nov 25 at 18:10