Let $T:X \to Y$ be a compact nonlinear operator (i.e., it is continuous and maps bounded sets into relatively compact sets) between two Banach spaces. Is it possible to approximate $T$ by a family of operators $T_n:X \to Y$ such that, if $x_n \rightharpoonup x$ in $X$ (weak convergence), then $T_n(x_n) \to T(x)$ in $Y$ (strongly)? The $T_n$ can be nonlinear.
If it helps $X=H^1_0(\Omega)$ and $Y$ is a subspace of $L^\infty(\Omega)$ of functions that are greater than or equal to a positive constant $c_0$.
The problem is getting the strong convergence for the $T_n$.