On equations of the form $\Delta u=f(x,u,Du)$

The Dirichlet problem is investigated for equations of the form $\Delta u=f(x,u,Du)$ in a bounded domain $\Omega$ in $\mathbf R^n$ with a $C^2$-boundary. This problem is studied in the Sobolev space $W^2_p(\Omega)$ with $p>n$. An exact condition is obtained for the growth of the function $f(x,u,\xi)$ with values in $L_p(\Omega)$ with respect to $\xi\in\mathbf R^n$, under which an a priori estimate of $\|u\|_\infty$ for the solution of the problem generates an estimate for $\|Du\|_\infty$. The theory of the solvability of such problems is studied, based on upper and lower solutions. Existence theorems are obtained.
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