Geodesic
On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with $L^1$-data
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 99-112 
A. A. Kovalevsky; F. Nicolosi. On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with $L^1$-data. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 99-112. http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a6/
@article{FPM_2006_12_4_a6,
     author = {A. A. Kovalevsky and F. Nicolosi},
     title = {On the sets of boundedness of solutions for a~class of degenerate nonlinear elliptic fourth-order equations with $L^1$-data},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {99--112},
     year = {2006},
     volume = {12},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a6/}
}
TY  - JOUR
AU  - A. A. Kovalevsky
AU  - F. Nicolosi
TI  - On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with $L^1$-data
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2006
SP  - 99
EP  - 112
VL  - 12
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a6/
LA  - ru
ID  - FPM_2006_12_4_a6
ER  - 
%0 Journal Article
%A A. A. Kovalevsky
%A F. Nicolosi
%T On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with $L^1$-data
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2006
%P 99-112
%V 12
%N 4
%U http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a6/
%G ru
%F FPM_2006_12_4_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened ellipticity condition and right-hand sides of the class $L^1$ depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular.

[1] Kovalevskii A. A., “Entropiinye resheniya zadachi Dirikhle dlya odnogo klassa nelineinykh ellipticheskikh uravnenii chetvertogo poryadka s $L^1$-pravymi chastyami”, Izv. RAN. Ser. mat., 65:2 (2001), 27–80 | MR | Zbl

[2] Skrypnik I. V., “Kvazilineinye ellipticheskie uravneniya vysshego poryadka s nepreryvnymi obobschennymi resheniyami”, Differents. uravn., 14 (1978), 786–795 | MR | Zbl

[3] Bénilan Ph., Boccardo L., Gallouët Th., Gariepy R., Pierre M., Vazquez J. L., “An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations”, Ann. Scu. Norm. Sup. Pisa Cl. Sci., 22 (1995), 241–273 | MR | Zbl

[4] Kovalevsky A., Nicolosi F., “Boundedness of solutions of variational inequalities with nonlinear degenerated elliptic operators of high order”, Appl. Anal., 65 (1997), 225–249 | DOI | MR | Zbl

[5] Kovalevsky A., Nicolosi F., “On Hölder continuity of solutions of equations and variational inequalities with degenerate nonlinear high order operators”, Problemi attuali dell'analisi e della fisica matematica (Atti del $2^0$ Simp. Int. dedicato alla memoria del Prof. Gaetano Fichera), Aracne Editrice, Roma, 2000, 205–220 | MR

[6] Kovalevsky A., Nicolosi F., “Solvability of Dirichlet problem for a class of degenerate nonlinear high-order equations with $L^1$-data”, Nonlinear Anal., Theory Methods Appl., 47 (2001), 435–446 | DOI | MR | Zbl

[7] Kovalevsky A., Nicolosi F., “Entropy solutions of Dirichlet problem for a class of degenerate anisotropic fourth-order equations with $L^1$-right-hand sides”, Nonlinear Anal., Theory Methods Appl., 50 (2002), 581–619 | DOI | MR | Zbl

[8] Kovalevsky A., Nicolosi F., “Existence of solutions of some degenerate nonlinear elliptic fourth-order equations with $L^1$-data”, Appl. Anal., 81 (2002), 905–914 | DOI | MR | Zbl

[9] Lions J. L., Quelques méthods de résolution des problèmes aux limites non linéaires Paris: Dunod, Gauthier-Villars, 1969 | MR

[10] Nicolosi F., Skrypnik I. V., “Nirenberg–Gagliardo interpolation inequality and regularity of solutions of nonlinear higher order equations”, Topol. Methods Nonlinear Anal., 7 (1996), 327–347 | MR | Zbl