Geodesic
Existence of solutions of certain quasilinear elliptic equations in~$\mathbb R^N$ without conditions at infinity
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 133-147.

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

The paper deals with conditions for the existence of solutions of the equations $$ -\sum_{i=1}^nD_iA_i(x,u,Du)+A_0(x,u)=f(x),\quad x\in\mathbb R^n, $$ considered in the whole space $\mathbb R^n$, $n\ge2$. The functions $A_i(x,u,\xi)$, $i=1,\dots,n$, $A_0(x,u)$, and $f(x)$ can arbitrarily grow as $|x|\to\infty$. These functions satisfy generalized conditions of the monotone operator theory in the arguments $u\in\mathbb R$ and $\xi\in\mathbb R^n$. We prove the existence theorem for a solution $u\in W_{\mathrm{loc}}^{1,p}(\mathbb R^n)$ under the condition $p>n$. 
@article{FPM_2006_12_4_a8,
     author = {G. I. Laptev},
     title = {Existence of solutions of certain quasilinear elliptic equations in~$\mathbb R^N$ without conditions at infinity},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {133--147},
     publisher = {mathdoc},
     volume = {12},
     number = {4},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a8/}
}
TY  - JOUR
AU  - G. I. Laptev
TI  - Existence of solutions of certain quasilinear elliptic equations in~$\mathbb R^N$ without conditions at infinity
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2006
SP  - 133
EP  - 147
VL  - 12
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a8/
LA  - ru
ID  - FPM_2006_12_4_a8
ER  - 
%0 Journal Article
%A G. I. Laptev
%T Existence of solutions of certain quasilinear elliptic equations in~$\mathbb R^N$ without conditions at infinity
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2006
%P 133-147
%V 12
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a8/
%G ru
%F FPM_2006_12_4_a8
G. I. Laptev. Existence of solutions of certain quasilinear elliptic equations in~$\mathbb R^N$ without conditions at infinity. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 133-147. http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a8/

[1] Gladkov A. L., “Zadacha Dirikhle dlya nekotorykh vyrozhdennykh ellipticheskikh uravnenii v neogranichennykh oblastyakh”, Differents. uravn., 29 (1993), 267–273 | MR | Zbl

[2] Dubinskii Yu. A., “Nelineinye parabolicheskie uravneniya vysokogo poryadka”, Itogi nauki i tekhn. Ser. Sovr. probl. matematiki, 37, VINITI, M., 1990, 89–166 | MR

[3] Kufner A., Fuchik S., Nelineinye differentsialnye uravneniya, Nauka, M., 1988 | MR

[4] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[5] Mitidieri E., Pokhozhaev S. I., “Apriornye otsenki i otsutstvie reshenii nelineinykh differentsialnykh uravnenii i neravenstv v chastnykh proizvodnykh.”, Trudy MIAN, 234, Nauka, M., 2001 | MR

[6] Skrypnik I. V., Metody issledovaniya nelineinykh ellipticheskikh granichnykh zadach, Nauka, M., 1990 | MR

[7] Brezis H., “Semilinear equation in $\mathbb R^N$ without condition at infinity”, Appl. Math. Optim., 12 (1984), 271–282 | DOI | MR | Zbl

[8] Kuzin I., Pohozaev S., Entire Solutions of Semilinear Elliptic Equations, Birkhäuser, 1997 | MR | Zbl

[9] Laptev G. I., “Solvability of quasilinear elliptic second order differential equations in $\mathbb R^n$ without condition at infinity”, Adv. Math. Research, 4 (2003), 1–18 | MR

[10] Leoni F., “Nonlinear elliptic equations in $\mathbb R^n$ with “absorbing” zero order terms”, Adv. Differential Equations, 5:4–5 (2000), 681–722 | MR | Zbl

[11] Oleinik O., Some Asymptotic Problems in the Theory of Partial Differential Equations, Cambridge Univ. Press, 1996 | MR | Zbl