Is there any good literature that mainly focus on quantitative estimates of solutions to variational problems? Take for example a generic 1-d variational problem such as wishing to choose a function $u$ in order to minimise the following functional, $$ \int_\Omega f(x,u(x),u'(x))dx,$$ where $\Omega \subset \mathbb{R}$. It is my experience that most books either focus on delving deep into specific problems or rather study general existence, uniqueness and regularity theorems. Say we could prove there exists a unique $C^2(\Omega)$ solution of the above minimisation problem so that the strong form Euler-Lagrange equations are satisfied. Often times if your problem is somewhat general, directly analysing the solution via the Euler-Lagrange equations is just not tractable.
I would be very interested in any references that try to obtain quantitative estimates on solutions via methods not associated with analysing the Euler-Lagrange equations directly. Even simple questions like does $u\to 0 $ as $x \to \infty$, or is $u$ (or $u'$) bounded, are of interest.
This question is intentionally general as what I'm really looking for is some possible references for such kind of problems rather than an answer to a specific problem.