Geodesic
A priori estimates, existence theorems, and the behavior at infinity of solutions of quasielliptic equations in $\mathbf{R}^n$
Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 437-452 
L. A. Bagirov. A priori estimates, existence theorems, and the behavior at infinity of solutions of quasielliptic equations in $\mathbf{R}^n$. Sbornik. Mathematics, Tome 38 (1981) no. 4, pp. 437-452. http://geodesic.mathdoc.fr/item/SM_1981_38_4_a0/
@article{SM_1981_38_4_a0,
     author = {L. A. Bagirov},
     title = {A~priori estimates, existence theorems, and the behavior at infinity of solutions of quasielliptic equations in $\mathbf{R}^n$},
     journal = {Sbornik. Mathematics},
     pages = {437--452},
     year = {1981},
     volume = {38},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_4_a0/}
}
TY  - JOUR
AU  - L. A. Bagirov
TI  - A priori estimates, existence theorems, and the behavior at infinity of solutions of quasielliptic equations in $\mathbf{R}^n$
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 437
EP  - 452
VL  - 38
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1981_38_4_a0/
LA  - en
ID  - SM_1981_38_4_a0
ER  - 
%0 Journal Article
%A L. A. Bagirov
%T A priori estimates, existence theorems, and the behavior at infinity of solutions of quasielliptic equations in $\mathbf{R}^n$
%J Sbornik. Mathematics
%D 1981
%P 437-452
%V 38
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1981_38_4_a0/
%G en
%F SM_1981_38_4_a0

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

The equation $$ A(x,D)u(x)=\sum_{\langle\alpha\cdot\theta\rangle\leqslant m}a_\alpha(x)D^\alpha u(x)=f(x),\qquad x\in\mathbf R^n, $$ is studied in this paper. Here $\theta=(\theta_1,\dots,\theta_n)$ is the index of quasihomogeneity of the operator $A$ and $\langle\alpha\cdot\theta\rangle=\alpha_1\theta_1+\dots+\alpha_n\theta_n$. The quasiellipticity condition $$ \biggl|\sum_{\langle\alpha\cdot\theta\rangle=m}a_\alpha(x)\xi^\alpha\biggr|\geqslant\delta\sum_{k=1}^n|\xi_k|^{m_k},\qquad\delta>0,\quad\xi\in\mathbf R^n,\quad x\in\mathbf R^n,\quad\frac{m_k}m=\theta_k^{-1}, $$ is assumed to hold. Theorems on the Noether property of $A$ in weighted spaces are proved under two types of conditions on the behavior of the coefficients $a_\alpha(x)$ at infinity. Bibliography: 18 titles.

[1] O. A. Besov, V. P. Ilin, L. D. Kudryavtsev, P. I. Lizorkin, S. M. Nikolskii, “Teoriya vlozhenii klassov differentsiruemykh funktsii mnogikh peremennykh”, Differentsialnye uravneniya s chastnymi proizvodnymi, izd-vo “Nauka”, Moskva, 1970, 38–63 | MR

[2] P. S. Filatov, “O differentsialnykh svoistvakh reshenii uravnenii kvaziellipticheskogo tipa na beskonechnosti”, Sib. matem. zh., XVI:2 (1975), 368–383

[3] P. S. Filatov, “O nëterovosti kvaziellipticheskikh uravnenii v $R^n$”, Differentsialnye uravneniya s chastnymi proizvodnymi, Trudy seminara S. L. Soboleva, no. 2, Novosibirsk, 1976, 129–138 | MR

[4] S. V. Uspenskii, “O differentsialnykh svoistvakh reshenii odnogo klassa psevdodifferentsialnykh uravnenii na beskonechnosti”, Sib. matem. zh., XIII:3 (1972), 665–678 | MR

[5] S. V. Uspenskii, “Differentsialnye svoistva reshenii odnogo klassa uravnenii v chastnykh proizvodnykh v neogranichennykh oblastyakh”, DAN SSSR, 196:1 (1971), 61–64 | MR | Zbl

[6] S. V. Uspenskii, “O predstavlenii funktsii, opredelyaemykh odnim klassom gippoellipticheskikh operatorov”, Trudy Matem. in-ta im. V. A. Steklova, CXVII, 1972, 292–299 | MR

[7] L. Nirenberg, H. F. Walker, “The null spaces of elliptic partial differential operators in $\mathbf R^n$”, J. Math. Anal. and Appl., 42:2 (1973), 271–301 | DOI | MR | Zbl

[8] L. A. Bagirov, V. A. Kondratev, “Ob ellipticheskikh uravneniyakh v $\mathbf R^n$”, Dif. uravneniya, XI:3 (1975), 498–504 | MR

[9] V. P. Mikhailov, “O povedenii na beskonechnosti odnogo klassa mnogochlenov”, Trudy Matem. in-ta im. V. A. Steklova, XCI, 1967, 59–80 | MR | Zbl

[10] M. S. Agranovich, M. I. Vishik, “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, XIX:3(117) (1964), 53–161 | MR | Zbl

[11] P. M. Blekher, “Ob operatorakh, zavisyaschikh meromorfno ot parametra”, Vestnik MGU, seriya matem. i mekh., 1969, no. 5, 30–36 | Zbl

[12] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Trudy Mosk. matem. ob-va, 16, 1967, 209–318

[13] L. A. Bagirov, Kraevye zadachi dlya obschikh ellipticheskikh i parabolicheskikh uravnenii v neogranichennykh oblastyakh, Kandidatskaya dissertatsiya, MGU, Moskva, 1969

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

[15] L. R. Volevich, “Lokalnye svoistva reshenii kvaziellipticheskikh sistem”, Matem. sb., 59(101) (dop.) (1962), 3–52 | MR | Zbl

[16] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii i deistviya nad nimi, vyp. 1, Fizmatgiz, Moskva, 1959

[17] L. A. Bagirov, “Ellipticheskie uravneniya v neogranichennykh oblastyakh”, Matem. sb., 86(128) (1971), 120–139 | MR

[18] S. V. Uspenskii, B. N. Chistyakov, “O vykhode na polinom pri stremlenii $|x|\to\infty$ reshenii odnogo klassa psevdodifferentsialnykh uravnenii”, Sib. matem. zh., XVI:5 (1975), 1053–1070 | MR