Geodesic
Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains
Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 855-866.

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

We study some nonexistence problems for the solutions of semilinear elliptic differential inequalities and systems of second order in conic domains. The proof is based on the trial function method developed by Mitidieri and Pokhozhaev without recourse to comparison theorems and to the maximum principle. 
@article{MZM_2002_71_6_a5,
     author = {G. G. Laptev},
     title = {Nonexistence of {Solutions} of {Elliptic} {Differential} {Inequalities} in {Conic} {Domains}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {855--866},
     publisher = {mathdoc},
     volume = {71},
     number = {6},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a5/}
}
TY  - JOUR
AU  - G. G. Laptev
TI  - Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains
JO  - Matematičeskie zametki
PY  - 2002
SP  - 855
EP  - 866
VL  - 71
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a5/
LA  - ru
ID  - MZM_2002_71_6_a5
ER  - 
%0 Journal Article
%A G. G. Laptev
%T Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains
%J Matematičeskie zametki
%D 2002
%P 855-866
%V 71
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a5/
%G ru
%F MZM_2002_71_6_a5
G. G. Laptev. Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains. Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 855-866. http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a5/

[1] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi i uglovymi tochkami”, Tr. MMO, 16, URSS, M., 1967, 209–292 | MR | Zbl

[2] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[3] Egnell H., “Positive solutions of semilinear equations in cones”, Trans. Amer. Math. Soc., 330 (1992), 191–201 | DOI | MR | Zbl

[4] Nguen Man Khung, “Ob otsutstvii polozhitelnykh reshenii nelineinykh ellipticheskikh uravnenii vtorogo poryadka v konicheskikh oblastyakh”, Differents. uravneniya, 34 (1998), 533–539 | MR

[5] Konkov A. A., “O neotritsatelnykh resheniyakh kvazilineinykh ellipticheskikh neravenstv”, Izv. RAN. Ser. matem., 63:2 (1999), 41–126 | MR | Zbl

[6] Kondratev V. A., Landis E. M., “O kachestvennykh svoistvakh reshenii odnogo nelineinogo uravneniya vtorogo poryadka”, Matem. sb., 135(177):3 (1988), 346–360 | MR | Zbl

[7] Pokhozhaev S. I., “Suschestvenno nelineinye emkosti, indutsirovannye differentsialnymi operatorami”, Dokl. RAN, 357 (1997), 592–594 | MR | Zbl

[8] Mitidieri E., Pokhozhaev S. I., “Otsutstvie globalnykh polozhitelnykh reshenii kvazilineinykh ellipticheskikh neravenstv”, Dokl. RAN, 359 (1998), 456–460 | MR | Zbl

[9] Mitidieri E., Pokhozhaev S. I., “Otsutstvie polozhitelnykh reshenii dlya kvazilineinykh ellipticheskikh zadach v $\mathbb R^N$”, Tr. MIAN, 227, Nauka, M., 1999, 192–222 | MR | Zbl

[10] Kurta V. V., Nekotorye voprosy kachestvennoi teorii nelineinykh differentsialnykh uravnenii vtorogo poryadka, Diss. ... d. f.-m. n., MIAN, M., 1994, 323 pp.

[11] Kurta V. V., “K voprosu ob otsutstvii tselykh polozhitelnykh reshenii u polulineinykh ellipticheskikh uravnenii”, UMN, 50:4 (1995), 131

[12] Kurta V. V., “Ob otsutstvii polozhitelnykh reshenii u polulineinykh ellipticheskikh uravnenii”, Tr. MIAN, 227, Nauka, M., 1999, 162–169 | MR | Zbl

[13] Mitidieri E., Pokhozhaev S. I., Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001

[14] Bandle C., Essen M., “On positive solutions of Emden equations in cone-like domains”, Arch. Rational Mech. Anal., 112 (1990), 319–338 | DOI | MR | Zbl

[15] Bandle C., “Positive solutions of Emden equations in a cone-like domains”, Progress in Nonlinear Differential Equations Appl., 7 (1992), 71–75 | MR | Zbl

[16] Laptev G. G., “Otsutstvie globalnykh polozhitelnykh reshenii sistem polulineinykh ellipticheskikh neravenstv v konusakh”, Izv. RAN. Ser. matem., 64:6 (2000), 107–124 | MR | Zbl

[17] Laptev G. G., “Some nonexistence results for higher-order evolution inequalities in cone-like domains”, Electron. Res. Announc. Amer. Math. Soc., 7 (2001), 87–93 | DOI | MR | Zbl

[18] Bidaut-Veron M.-F., Pohozaev S., “Nonexistence results and estimates for some nonlinear elliptic problems”, J. Anal. Math., 84 (2001), 1–49 | DOI | MR | Zbl

[19] Kartsatos A. G., Kurta V. V., “Nonexistence theorems for weak solutions of quasilinear elliptic equations”, Abstract Appl. Anal., 2000, no. 6, 163–189

[20] Brezis H., Cabré X., “Some simple nonlinear PDE's without solutions”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1 (1998), 223–262 | MR | Zbl

[21] Galakhov E., “Some nonexistence results for quasilinear elliptic problems”, J. Math. Anal. Appl., 252 (2000), 256–277 | DOI | MR | Zbl

[22] Pokhozhaev S. I., Tesei A., “Polnoe razrushenie reshenii nelineinykh ellipticheskikh neravenstv”, Differents. uravneniya, 37 (2001), 521–528 | MR | Zbl

[23] Laptev G. G., “Ob otsutstvii reshenii odnogo klassa singulyarnykh polulineinykh differentsialnykh neravenstv”, Tr. MIAN, 232, Nauka, M., 2001, 223–235 | MR

[24] Mitidieri E., Pokhozhaev S. I., “Nekotorye obobscheniya teoremy Bernshteina”, Differents. uravneniya, 38 (2002) (to appear) | MR