Existence and uniqueness of solutions for a class of degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro

Abstract


In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to 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}_1(x) \mathcal{A}_j(x, u, {\nabla}u){\bigr]}+ b(x,u,{\nabla}u)\, {\omega}_2(x)\\ & = f_0(x) - \sum_{j=1}^nD_jf_j(x), \ \ {\rm in } \ \ {\Omega} \end{align} in the setting of the weighted Sobolev spaces.

Keywords


Degenerate nonlinear elliptic equations; weighted Sobolev spaces

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References


Cavalheiro, A. C., Existence and uniqueness of solutions for some degenerate nonlinear Dirichlet problems, J. Appl. Anal. 19 (2013), 41-54.

Cavalheiro, A. C., Existence results for Dirichlet problems with degenerated p-Laplacian and p-Biharmonic operators, Appl. Math. E-Notes 13 (2013), 234-242.

Chipot, M., Elliptic Equations: An Introductory Course, Birkhauser, Berlin, 2009.

Drabek, P., Kufner, A., Nicolosi, F., Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, Berlin, 1997.

Fucik, S., John, O., Kufner, A., Function Spaces, Noordhoff International Publ., Leyden, 1977.

Garcia-Cuerva, J., Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.

Gilbarg, D., Trudinger, N. S., Elliptic Partial Equations of Second Order, 2nd Ed., Springer, New York, 1983.

Heinonen, J., Kilpelainen, T., Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Inc., New York, 1993.

Kufner, A., Weighted Sobolev Spaces, John Wiley & Sons, 1985.

Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.

Talbi, M., Tsouli, N., On the spectrum of the weighted p-Biharmonic operator with weight, Mediterr. J. Math. 4 (2007), 73-86.

Torchinsky, A., Real-Variable Methods in Harmonic Analysis, Academic Press, San Diego, 1986.

Turesson, B. O., Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 2000.

Zeidler, E., Nonlinear Functional Analysis and Its Applications. Vol. I, Springer-Verlag, New York, 1990.

Zeidler, E., Nonlinear Functional Analysis and Its Applications. Vol. II/B, Springer-Verlag, New York, 1990.




DOI: http://dx.doi.org/10.17951/a.2016.70.2.9
Date of publication: 2016-12-24 22:42:00
Date of submission: 2016-12-22 23:18:14


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