Geodesic
Asymptotics of the solution of the nonlinear Dirichlet problem having a strong singularity near a corner point
Izvestiya. Mathematics, Tome 25 (1985) no. 3, pp. 531-550 
S. A. Nazarov; K. I. Pileckas. Asymptotics of the solution of the nonlinear Dirichlet problem having a strong singularity near a corner point. Izvestiya. Mathematics, Tome 25 (1985) no. 3, pp. 531-550. http://geodesic.mathdoc.fr/item/IM2_1985_25_3_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The asymptotics of the solutions of the Dirichlet problem for the equation $$ -\Delta u(x)+u(x)^{2k+1}=f(x),\qquad x\in\Omega, $$ is studied in a plane domain $\Omega$ with a corner point of angle $\alpha$. The asymptotics of a solution of this problem is constructed in the case where the right side $f$ has a strong singularity near the corner point. Bibliography: 12 titles.

[1] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Moskovskogo matem. obschestva, 16, 1967, 209–292

[2] Kondratev V. A., “Asimptotika reshenii uravnenii Nave–Stoksa v okrestnosti uglovoi tochki granitsy”, Prikl. matematika i mekhanika, 31 (1967), 119–123

[3] Mazya V. G., Plamenevskii B. A., “O povedenii reshenii kvazilineinykh ellipticheskikh kraevykh zadach v okrestnosti konicheskoi tochki”, Zapiski nauchnykh seminarov Leningr. otd. Matem. in-ta AN SSSR, 38, Nauka, L., 1973, 94–97 | MR | Zbl

[4] Mazya V. G., Plamenevskii B. A., “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblastyakh s konicheskimi tochkami”, Math. Nachr., 76 (1977), 29–60 | DOI

[5] Mazya V. G., Morozov N. F., Plamenevskii B. A., “O nelineinom izgibe plastiny s treschinoi”, Differentsialnye i integralnye uravneniya, kraevye zadachi, Metsniereba, Tbilisi, 1979, 145–163 | MR

[6] Nazarov S. A., “Asimptotika vblizi uglovoi tochki resheniya odnogo nelineinogo uravneniya”, Matem. zametki, 31 (1982), 411–420 | MR | Zbl

[7] Mazya V. G., Plamenevskii B. A., “O resheniyakh nelineinoi zadachi Dirikhle s silnymi osobennostyami v uglovykh tochkakh”, Uspekhi matem. nauk, 37 (1982), 101

[8] Vishik M. P., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi matem. nauk, 12 (1957), 3–122 | Zbl

[9] Nazarov S. A., “Asimptotika po malomu parametru resheniya ellipticheskoi po Agranovichu–Vishiku kraevoi zadachi v oblasti s konicheskoi tochkoi”, Problemy matematicheskogo analiza, no. 7, LGU, L., 1979, 146–167 | MR

[10] Nazarov S. A., “Metod Vishika–Lyusternika dlya ellipticheskikh kraevykh zadach v oblastyakh s konicheskimi tochkami. I: Zadacha v konuse”, Sibirskii matem. zhurnal, 22:4 (1981), 142–163 | MR | Zbl

[11] Gaevskii X., Grëger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978, 336 pp. | MR

[12] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973, 576 pp. | MR