Convergence of solution of PDE as forcing term varies.

Question

Let $L$ be an elliptic operator.

We have the PDE

$Lf(x; y) = u(x; y)$

on some domain $\Omega$ with Dirichlet boundary condition. Suppose we know that each PDE admits unique solution.

$u$ is a sequence of $C^1$ function, known to converge (at least pointwise), as y converges to infinity. What result do we have about convergence of $f(x; y)$?