Reference Request: Estimates for Euler Lagrange PDE

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I am currently working on an applied problem where I need estimates for the solution of an Euler-Lagrange equation for a functional $$I(u)=\int L dp $$ In N-dimensional euclidean space. I have looked through Evan's book but didn't find what I need.

I am most interested in $L^2/H^2/L^\infty$ estimates which depend at least partially on the data/form of L.

I realize this question is quite broad so really I'll appreciate any pointers.

Kindly Afc