In some existence proof for the stationary incompressible Navier-Stokes equations I came across the following Lemma:
For $\Omega \subset \mathbb{R}^n$, $\psi _k \in C_c^\infty(\Omega)$ for $k=1,...,m$ with
\begin{equation} (\psi _k, \psi _l)= \delta_{kl} \end{equation}
we consider the system
\begin{equation} \nu (\nabla w, \nabla \psi _k) = F_k(\xi),\ \ \ k=1,...,m \ \ \ (1) \end{equation}
\begin{equation} w = \sum _{k=1}^m \xi _k \psi _k \ \ \ (2) \end{equation}
with $\nu > 0,$ $\xi = (\xi_1,...,\xi_m)\in \mathbb{R}^m$ and $F:\mathbb{R}^m \rightarrow \mathbb{R}^m$ continuous. If there are $c>0$ and $0< a < \nu$ satisfying
\begin{equation} F(\xi)\cdot \xi \leq c\|w\|_{1,2} + a \|w\|_{1,2}^2 \ \ \ (3) \end{equation}
then $(1)$, $(2)$ admits a solution $\xi \in \mathbb{R}^m$.
Now it further says that condition (3) may be generalized, but unfortunately no references are mentioned. In particular, I would be interested in a version of the Lemma where $a < \nu$ can be relaxed or with some higher order terms on the right hand side, but I do not what to look for. Can somebody tell me where I might find results like this?