Existence and uniqueness theorems for $A$-regular generalized solutions of the first boundary-value problem for $(A,\vec0)$-elliptic equations

For second-order quasilinear degenerate elliptic equations, having the structure of $(A,\vec0)$-elliptic equations in a bounded domain $\Omega\subset R^n$, $n\ge2$, one establishes theorems of existence and uniqueness for the generalized solutions of the first boundary-value problem, bounded together with their $A$-derivatives of first order and also of first and second order. The case of linear second-order $(A,\vec0)$-elliptic equations are separately considered.