Fredholm alternative vs variational methods I was reading an exercise in a lecture notes of variational methods in order to show that if $\rho: \overline{\Omega} \rightarrow \mathbb{R}$ is continuous in a bounded domain $\Omega \subset \mathbb{R}^N$, then a necessary condition so that $-\Delta u = \lambda_j u + \rho(x)$ has a weak solution in $H_0^1(\Omega)$ is that $\int_{\Omega} \rho(x) v dx = 0$ for all $v \in \ \text{ker} \ (\Delta + \lambda_j Id)$.
I did this exercise with Fredholm alternative (a tool of functional analysis), but this exercise in a lecture notes of variational methods led me to think in a comparation between Fredholm alternative and variational methods, so my question is: are there some advantages in Fredholm alternative compared to variational methods and vice versa to solve a PDE problem? If possible, give me examples, please.
Thanks in advance!
$\textbf{P.S.:}$ by "variational methods" I mean the study of find weak solutions of a PDE problem associating a functional to this problem and find a critical point for this functional that is a weak solution for the problem.

You cannot analyze the solvability of an equation like
$$
-\Delta u + b\cdot u = f
$$
by energy minimization methods. The bilinear form associated to weak solutions of the above equation is not symmetric. If studying critical points of a quadratic energy functional the resulting bilinear form is inevitably symmetric. So these kind of variational methods do not work for such equations.
