A priori estimates and smoothness of solutions of a system of quasi-linear equations that is elliptic in the Douglis–Nirenberg sense

We study a Douglis–Nirenberg elliptic system of quasi-linear equations. We solve the problem of the limiting admissible rate of growth of the non-linear terms of the system with respect to their arguments consistent with the possibility of obtaining estimates of the derivatives of a solution in terms of its maximum absolute value. The restrictions on the smoothness of the non-linear terms are minimal and the results are sharp. We construct an example that shows the optimality of the upper bound for the exponent of growth. A priori $L_p$-estimates are obtained both inside the domain for solutions belonging to certain Sobolev spaces. We obtain estimates of the Hölder norms of the derivatives of a solutions. We prove a theorem on a removable isolated singularity of bounded solutions of general elliptic systems of quasi-linear equation. All results are new, even for a single second-order equation.