What is needed is an a-priori $L_\infty$ bound for the solution $v_k$. If you know such an a-priori bound, you can modify $\zeta$ outside of this bound, and you can assume without of generality that $\zeta(u)=0$ for large $|u|$.

Then the finite-dimensional equation is of the form
$$Av+F(v)=0$$
where $A$ is linear and positive definite, and $F$ is continuous and bounded. In particular, $A^{-1}$ exists, and the equation thus is equivalent to
$$v=-A^{-1}F(v)\text.$$
The range of the map $G=-A^{-1}F$ is contained in some ball. In particular, $G$ maps this ball into itself, and so Brouwer's fixed point theorem implies that $G$ has a fixed point which thus is a solution of the finite-dimensional equation.