Geodesic
On equations of the form $\Delta u=f(x,u,Du)$
Sbornik. Mathematics, Tome 41 (1982) no. 2, pp. 269-280 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] H. Amann, M. G. Crandall, “On some existence theoremes for semi-linear elliptic equations”, Ind. Univ. Math. J., 27:5 (1978), 779–790 | DOI | MR | Zbl

[2] I. L. Kazdan, R. I. Kramer, “Invariant criteria for existence of solutions to second-order quasilinear elliptic equations”, Comm. Pure Appl. Math., 31:5 (1978), 619–645 | DOI | MR | Zbl

[3] S. N. Bernshtein, Ob uravneniyakh variatsionnogo ischisleniya, Sobranie sochinenii, t. III, izd-vo AN SSSR, Moskva, 1960

[4] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, izd-vo “Nauka”, Moskva, 1973 | MR

[5] A. D. Aleksandrov, “Usloviya edinstvennosti i otsenki resheniya zadachi Dirikhle”, Vestnik LGU, seriya matem., mekh. i astronomii, 1963, no. 13, vyp. 3, 5–29 | Zbl

[6] A. Friedman, Partial Differential Equations, Academic Press, New York, 1969 | MR

[7] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, izd-vo SO AN SSSR, Novosibirsk, 1962