On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

Cavalheiro, Albo Carlos

doi:10.2478/amsil-2024-0024

Abstract

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations $\begin{matrix} - div 𝒜 (x, u, Δ u) ω_{1} + ℬ (x, u, \nabla u) ν_{1}] + ℋ (x, u, \nabla u) ν_{2} + {u|}^{p - 2} u ω_{2} \\ - \sum_{i, j = 1}^{n} D_{j} (a_{ij} (x) D_{i} u (x)) = f_{0} (x) - \sum_{j = 1}^{n} D_{j} f_{j} (x) in Ω, \\ u (x) = 0 on \partial Ω, \end{matrix}$ \begin{array}{*{20}{c}}{ - \;{\rm{div}}\left[ {\mathcal{A}\left( {x,u,\Delta u} \right)\;{\omega _1} + {\mathcal{B}}\left( {x,u,\nabla u} \right){\nu _1}} \right] + \mathcal{H}\left( {x,u,\nabla u} \right){\nu _2} + {{\left| u \right|}^{p - 2}}u\;{\omega _2}}\\{ - \sum\limits_{i,j = 1}^n {{D_j}\left( {{a_{ij}}\left( x \right){D_i}u\left( x \right)} \right)} = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)} \;\;\;\;\;{\rm{in}}\;\;\Omega ,}\\{u\left( x \right) = 0\;\;\;\;\;{\rm{on}}\;\partial \Omega ,}\end{array} in the setting of the weighted Sobolev spaces.

References

A. Björn, J. Björn, and A. Christensen, Poincaré inequalities and Ap weights on bowties, arXiv preprint, 2022. Available at arXiv: 2202.07491.
Search in Google Scholar Back to article
D. Bresch, J. Lemoine, and F. Guíllen-Gonzalez, A note on a degenerate elliptic equation with applications for lakes and seas, Electron. J. Differential Equations (2004), No. 42, 13 pp.
Search in Google Scholar Back to article
A.C. Cavalheiro, Existence of solutions for Dirichlet problem of some degenerate quasilinear elliptic equations, Complex Var. Elliptic Equ. 53 (2008), no. 2, 185–194.
Search in Google Scholar Back to article
A.C. Cavalheiro, Existence results for Dirichlet problems with degenerated p-Laplacian, Opuscula Math. 33 (2013), no. 3, 439–453.
Search in Google Scholar Back to article
A.C. Cavalheiro, Weighted Sobolev Spaces and Degenerate Elliptic Equations, Cambridge Scholars Publishing, Newcastle upon Tyne, UK, 2023.
Search in Google Scholar Back to article
M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Verlag, Berlin, 2009.
Search in Google Scholar Back to article
M. Colombo, Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations. With Applications to the Vlasov-Poisson and Semigeostrophic Systems, Edizioni della Normale, Pisa, 2017.
Search in Google Scholar Back to article
P. Drábek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter & Co., Berlin, 1997.
Search in Google Scholar Back to article
E.B. Fabes, C.E. Kenig, and R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116.
Search in Google Scholar Back to article
J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Publishing Co., Amsterdam, 1985.
Search in Google Scholar Back to article
J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, The Clarendon Press, Oxford University Press, New York, 1993.
Search in Google Scholar Back to article
A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, Inc., New York, 1985.
Search in Google Scholar Back to article
A. Kufner, O. John, and S. Fučik, Function Spaces, Noordhoff International Publishing, Leiden; Academia, Prague, 1977.
Search in Google Scholar Back to article
A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin. 25 (1984), no. 3, 537–554.
Search in Google Scholar Back to article
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.
Search in Google Scholar Back to article
B. Opic and A. Kufner, Hardy-Type Inequalities, Longman Scientific & Technical, Harlow, 1990.
Search in Google Scholar Back to article
A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, Inc., San Diego, 1986.
Search in Google Scholar Back to article
B.O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer-Verlag, Berlin, 2000.
Search in Google Scholar Back to article
E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. I, Springer-Verlag, Berlin, 1986.
Search in Google Scholar Back to article
E. Zeidler, Nonlinear Functional Analysis and its Applications, vol.II/B, Springer-Verlag, Berlin, 1990.
Search in Google Scholar Back to article

On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

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