I'm interested in an estimate of the type $$||u_h||_\infty \leq C ||f||_2$$ for $u_h$ solving $-\Delta u_h+c_0u_h = f$ ($u_h$ belongs to standard piecewise affine finite element space that are zero on the boundary), on the domain of interest $\Omega$, which for the sake of simplicity is convex and can be exactly approximated by the family of meshes. Here $f \in L^2$ (or in $L^p$ with suitably large $p$), and $c_0$ is non negative and bounded.
Do you know one, for $C$ not depending on $h$ and $c_0$?
In the continuous version, a similar stability estimate is available for general semilinear problems, see for instance p. $358$ of this book. In the proof, a truncation of $u$ is considered. This argument won't work for my finite dimensional version, as truncating $u_h$ may take us out of the finite element space (plus other technical reasons).
I'd appreciate a discrete version, thanks in advance!
It seems that the answer in general is yes, and one appeals to the finite element counterpart of elliptic regularity theory. Theorem 42 from https://arxiv.org/abs/2004.09341 shows that your bound holds for standard finite element approximation in the $C^\alpha$ semi norm (which is an upper bound on the supremum norm for functions with zero boundary value), when $f\in L^p$, $p>n/2$. Another example holds for convex domains, where a strong formulation of the problem is considered (see e.g. https://arxiv.org/abs/1911.05407 remark 3.3) - which is a non standard finite element method.
A simple case where the proof is easy is for standard finite elements in 1D. In this case by sobolev embeddings one has that $\|u\|_\infty \le C\|u\|_{H^1}$ for any $H^1$ function. Since also one immediately finds from coercivity that the finite element solution satisfies $$\|u_h\|_{H^1}\le C\|f\|_2, $$ (here C can be independent of $c_0$ if $c_0\ge 0$) and so one obtains the supremum bound.