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Keywords:
degenerate nolinear elliptic equations; weighted Sobolev spaces
degenerate nolinear elliptic equations; weighted Sobolev spaces
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.
References:
[1] Cavalheiro, A.C.: Existence results for Dirichlet problems with degenerate p-Laplacian. Opuscula Math. 33 (2013), no. 3, 439–453. DOI 10.7494/OpMath.2013.33.3.439 | MR 3046406
[2] Cavalheiro, A.C.: Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems. J. Appl. Anal. 19 (2013), 41–54. DOI 10.1515/jaa-2013-0003 | MR 3069764 | Zbl 1278.35086
[3] Chipot, M.: Elliptic Equations: An Introductory Course. Birkhäuser, Berlin, 2009. MR 2494977 | Zbl 1171.35003
[4] Drábek, P., Kufner, A., Nicolosi, F.: Quasilinear Elliptic Equations with Degenerations and Singularities. Walter de Gruyter, Berlin, 1997. MR 1460729 | Zbl 0894.35002
[5] Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations (1982), 77–116. DOI 10.1080/03605308208820218 | MR 0643158 | Zbl 0498.35042
[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
[7] Garcia-Cuerva, J., de Francia, J.L. Rubio: Weighted Norm Inequalities and Related Topics. North-Holland Math. Stud. 116 (1985). MR 0807149
[8] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Equations of Second Order. second ed., Springer, New York, 1983. MR 0737190
[9] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs, Clarendon Press, 1993. MR 1207810 | Zbl 0780.31001
[10] Kufner, A.: Weighted Sobolev Spaces. John Wiley and Sons, 1985. MR 0802206 | Zbl 0579.35021
[11] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–226. DOI 10.1090/S0002-9947-1972-0293384-6 | MR 0293384 | Zbl 0236.26016
[12] Talbi, M., Tsouli, N.: On the spectrum of the weighted p-Biharmonic operator with weight. Mediterranean J. Math. 4 (2007), 73–86. DOI 10.1007/s00009-007-0104-3 | MR 2310704 | Zbl 1150.35072
[13] Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Academic Press, São Diego, 1986. MR 0869816 | Zbl 0621.42001
[14] Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Math., vol. 1736, Springer-Verlag, 2000. MR 1774162 | Zbl 0949.31006
[15] Zeidler, E.: Nonlinear Functional Analysis and its Applications. vol. II/B, Springer-Verlag, 1990. MR 1033498 | Zbl 0684.47029
[16] Zeidler, E.: Nonlinear Functional Analysis and its Applications. vol. I, Springer-Verlag, 1990. Zbl 0684.47029
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