An alternative.
The given problem is equivalent to the minimization problem \begin{align} \mathcal{J}(v_k)= \min_{v\in \mathcal{V}_k} \mathcal{J}(v)\quad \text{with}\quad \mathcal{J}(v):= \frac12 \int_\Omega|\nabla v|^2 dx + \int_\Omega G(v)d x + \int_\Omega fvd x \end{align} and we define the function $G(v)= \int_0^v\zeta(\tau)\tau d \tau $. Note that $G$ is non-negative since $\zeta(\tau)\geq 0$. Using the Poincar'e-Friedrichs inequality we find that $\mathcal{J}(v)\to \infty$, as $\|v\|_{L^2(\Omega)} \to \infty$ and $v\in \mathcal{V}_k$. Which implies the existence of a minimizer $v_k\in \mathcal{V}_k$ of \mathcal{J}, since we are in finite dimension space.