Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

Albo Carlos Cavalheiro

doi:10.2478/mjpaa-2019-0012

.blurhash-client-img { display: none !important; }

Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

Moroccan Journal of Pure and Applied Analysis

Volume 5 (2019): Issue 2 (December 2019)

By: Albo Carlos Cavalheiro

Open Access

|Jan 2020

Abstract

In this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations

${\begin{matrix} - div [𝒜 (x, \nabla u) ω_{1} + 𝒝 (x, u, \nabla u) ω_{2}] = f_{0} (x) - \sum_{j = 1}^{n} D_{j} f_{j} (x) in Ω, \\ u (x) = 0 on \partial Ω, \end{matrix}$ \left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right.

in the setting of the weighted Sobolev spaces.

References

[1] D. Bresch, J. Lemoine, F. Guíllen-Gonzalez, A note on a degenerate elliptic equation with applications for lake and seas, Electron. J. Differential Equations, vol. 2004 (2004), No. 42, 1-13.
Search in Google Scholar Back to article
[2] A.C.Cavalheiro, Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems, J. Appl. Anal., 19 (2013), 41-54.10.1515/jaa-2013-0003
Search in Google Scholar Back to article
[3] A.C.Cavalheiro, Existence results for Dirichlet problems with degenerate p-Laplacian, Opuscula Math., 33, no 3 (2013), 439-453.10.7494/OpMath.2013.33.3.439
Search in Google Scholar Back to article
[4] A.C.Cavalheiro, Topics on Degenerate Elliptic Equations, Lambert Academic Publishing, Germany (2018).
Search in Google Scholar Back to article
[5] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser, Berlin (2009).10.1007/978-3-7643-9982-5
Search in Google Scholar Back to article
[6] M. Colombo, Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations: With Applications to the Vlasov-Poisson and Semigeostrophic Systems, Publications on the Scuola Normale Superiore Pisa, 22, Pisa (2017).10.1007/978-88-7642-607-0_8
Search in Google Scholar Back to article
[7] P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, Berlin (1997)10.1515/9783110804775
Search in Google Scholar Back to article
[8] E. Fabes, C. Kenig, R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116.10.1080/03605308208820218
Search in Google Scholar Back to article
[9] S. Fučik, O. John and A. Kufner, Function Spaces, Noordhoff International Publ., Leyden, (1977).
Search in Google Scholar Back to article
[10] J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies 116, (1985).
Search in Google Scholar Back to article
[11] D.Gilbarg and N.S. Trudinger, Elliptic Partial Equations of Second Order, 2nd Ed., Springer, New York (1983).
Search in Google Scholar Back to article
[12] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monographs, Clarendon Press, (1993).
Search in Google Scholar Back to article
[13] A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, Germany (1985).
Search in Google Scholar Back to article
[14] A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin., 25 (1984), 537 - 554.
Search in Google Scholar Back to article
[15] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Am. Math. Soc. 165 (1972), 207-226.10.1090/S0002-9947-1972-0293384-6
Search in Google Scholar Back to article
[16] M. Talbi and N. Tsouli, On the spectrum of the weighted p-Biharmonic operator with weight, Mediterr. J.Math., 4 (2007), 73-86.10.1007/s00009-007-0104-3
Search in Google Scholar Back to article
[17] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, (1986).
Search in Google Scholar Back to article
[18] B.O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Math., vol. 1736, Springer-Verlag, (2000).10.1007/BFb0103908
Search in Google Scholar Back to article
[19] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol.I, Springer-Verlag, Berlin (1990).10.1007/978-1-4612-0981-2
Search in Google Scholar Back to article
[20] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol.II/B, Springer-Verlag, Berlin (1990).10.1007/978-1-4612-0981-2
Search in Google Scholar Back to article