Seeking corresponding functional to show existence of weak solution to PDE $$-\Delta  u = \lambda u \space, \space\space\space\ x∈Ω$$
$$u(x) = 0\space, \space \space\space\space\space x∈\partial Ω$$
What functional would form a good correspondence, and whose minimization could show that the above function, $u$, has a weak solution in $H^1_0(\Omega)$? Is this analogous solving the case of:
$$-\Delta  u = f(x) \space, \space\space\space\ x∈Ω$$
$$u(x) = 0\space, \space \space\space\space\space x∈\partial Ω$$
taking $f(x)=\lambda u$, or is the there a better way to solve this specific case?
 A: If $\lambda\leq 0$ and is known, then the equation is nothing but a regular Poisson equation with an $L^2$-term.

If $\lambda >0$ is is unknown, then this is an eigenvalue problem, the link toypajme's comment is pretty nice and readable. Here I just add a bit more about motivation. Normally for eigenvalue problem, we are minimizing the Rayleigh quotient:
$$
F(u) = \frac{\| \nabla u\|_{L^2(\Omega)}^2}{\| u\|_{L^2(\Omega)}^2},\tag{1}
$$
whereas 
$$\int_\Omega u w_i = 0\quad \text{for eigenfunction } w_i, i=1,\ldots,n-1,$$
and the minimizer $u = w_n$ is another eigenfunction while $F(w_n)$ is the $n$-th eigenvalue. Notice problem (1) can be written as the same constraint minimization problem in the link in the toypajme's comment.

If $\lambda = \omega^2$ is known and is not one of the eigenvalues for the operator $-\Delta$, also we have a right hand side $f$, then this is a Helmholtz equation, and it weak formulation 
$$
\int_{\Omega}\nabla u\cdot \nabla v - \omega^2 \int_{\Omega}uv = \int_\Omega fv
$$
has a unique solution in $H^1_0$ by Fredholm alternative.
