  A general class of strongly coupled elliptic systems with quadratic growth in
gradients is considered and the existence of their strong solutions is
established. The results greatly improve those in a recent paper \cite{dleJFA}
as the systems can be either degenerate or singular when their solutions become
unbounded. A unified proof for both cases is presented. Most importantly, the
VMO assumption in \cite{dleJFA} will be replaced by a much versatile one thanks
to a new local weighted Gagliardo-Nirenberg involving BMO norms. Examples in
physical models will be provided.
