New a priori estimates for $q$-nonlinear elliptic systems with strong nonlinearities in the gradient, $1<q<2$

We consider $q$-nonlinear nondiagonal elliptic systems, $1<q<2$, with strong nonlinear terms in the gradient. Under a smallness condition on the gradient of a solution in the Morry space $L^{q,n-q}$, we estimate $L^p$-norm of the gradient, $p>q$, and the Hölder norm of the solution for the case $n=2$. An abstract theorem on “quasireverse Hölder inequalities” proved by the author earlier is essencially used.