| Title:
|
Existence and uniqueness of solutions for some degenerate nonlinear elliptic equations (English) |
| Author:
|
Cavalheiro, Albo Carlos |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
50 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
51-63 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article we are interested in the existence and uniqueness of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations
\begin{align*}{\Delta }(v(x)\, {\vert {\Delta }u\vert }^{p-2}{\Delta }u) &-\sum _{j=1}^n D_j{\bigl [}{\omega }(x) {\mathcal{A}}_j(x, u, {\nabla }u){\bigr ]}\\ =&\ f_0(x) - \sum _{j=1}^nD_jf_j(x)\,, \quad \mbox {in}\quad {\Omega }\end{align*}
in the setting of the weighted Sobolev spaces. (English) |
| Keyword:
|
degenerate nolinear elliptic equations |
| Keyword:
|
weighted Sobolev spaces |
| MSC:
|
35J60 |
| MSC:
|
35J70 |
| idZBL:
|
Zbl 06391565 |
| idMR:
|
MR3194768 |
| DOI:
|
10.5817/AM2014-1-51 |
| . |
| Date available:
|
2014-04-04T07:19:45Z |
| Last updated:
|
2015-03-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143719 |
| . |
| Reference:
|
[1] Cavalheiro, A.C.: Existence results for Dirichlet problems with degenerate p-Laplacian.Opuscula Math. 33 (2013), no. 3, 439–453. MR 3046406, 10.7494/OpMath.2013.33.3.439 |
| Reference:
|
[2] Cavalheiro, A.C.: Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems.J. Appl. Anal. 19 (2013), 41–54. Zbl 1278.35086, MR 3069764, 10.1515/jaa-2013-0003 |
| Reference:
|
[3] Chipot, M.: Elliptic Equations: An Introductory Course.Birkhäuser, Berlin, 2009. Zbl 1171.35003, MR 2494977 |
| Reference:
|
[4] Drábek, P., Kufner, A., Nicolosi, F.: Quasilinear Elliptic Equations with Degenerations and Singularities.Walter de Gruyter, Berlin, 1997. Zbl 0894.35002, MR 1460729 |
| Reference:
|
[5] Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations.Comm. Partial Differential Equations (1982), 77–116. Zbl 0498.35042, MR 0643158, 10.1080/03605308208820218 |
| Reference:
|
[6] Fučik, S., John, O., Kufner, A.: Function spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis.Noordhoff International Publishing, Leyden; Academia, Prague, 1977. MR 0482102 |
| Reference:
|
[7] Garcia-Cuerva, J., de Francia, J.L. Rubio: Weighted Norm Inequalities and Related Topics.North-Holland Math. Stud. 116 (1985). MR 0807149 |
| Reference:
|
[8] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Equations of Second Order.second ed., Springer, New York, 1983. MR 0737190 |
| Reference:
|
[9] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations.Oxford Math. Monographs, Clarendon Press, 1993. Zbl 0780.31001, MR 1207810 |
| Reference:
|
[10] Kufner, A.: Weighted Sobolev Spaces.John Wiley and Sons, 1985. Zbl 0579.35021, MR 0802206 |
| Reference:
|
[11] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function.Trans. Amer. Math. Soc. 165 (1972), 207–226. Zbl 0236.26016, MR 0293384, 10.1090/S0002-9947-1972-0293384-6 |
| Reference:
|
[12] Talbi, M., Tsouli, N.: On the spectrum of the weighted p-Biharmonic operator with weight.Mediterranean J. Math. 4 (2007), 73–86. Zbl 1150.35072, MR 2310704, 10.1007/s00009-007-0104-3 |
| Reference:
|
[13] Torchinsky, A.: Real-Variable Methods in Harmonic Analysis.Academic Press, São Diego, 1986. Zbl 0621.42001, MR 0869816 |
| Reference:
|
[14] Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces.Lecture Notes in Math., vol. 1736, Springer-Verlag, 2000. Zbl 0949.31006, MR 1774162 |
| Reference:
|
[15] Zeidler, E.: Nonlinear Functional Analysis and its Applications.vol. II/B, Springer-Verlag, 1990. Zbl 0684.47029, MR 1033498 |
| Reference:
|
[16] Zeidler, E.: Nonlinear Functional Analysis and its Applications.vol. I, Springer-Verlag, 1990. Zbl 0684.47029 |
| . |
