Geodesic
On equations of the form $\Delta u=f(x,u,Du)$
Sbornik. Mathematics, Tome 41 (1982) no. 2, pp. 269-280

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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. Bibliography: 7 titles. 
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     author = {S. I. Pokhozhaev},
     title = {On equations of the form $\Delta u=f(x,u,Du)$},
     journal = {Sbornik. Mathematics},
     pages = {269--280},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {1982},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1982_41_2_a6/}
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S. I. Pokhozhaev. On equations of the form $\Delta u=f(x,u,Du)$. Sbornik. Mathematics, Tome 41 (1982) no. 2, pp. 269-280. http://geodesic.mathdoc.fr/item/SM_1982_41_2_a6/