Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data

Document Type : Research Paper

Authors

Laboratory LMACS, FST of Beni Mellal, BP 523, 23000, Sultan Moulay Slimane University, Morocco

Abstract

In this paper, we study the existence and uniqueness of weak solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic problems
\begin{equation*}
-{\rm{div}}\Big[ \omega_{1}\mathcal{A}(x,\nabla u)+\nu_{2}\mathcal{B}(x,u,\nabla u)\Big]+ \nu_{1}\mathcal{C}(x,u)+ \omega_{2}\vert u\vert^{p-2}u=f-{\rm{div}}F,
\end{equation*}
where $1 < p < \infty$, $\omega_{1}$, $\nu_{2}$, $\nu_{1}$ and $\omega_{2}$ are $A_p$-weight functions, and $\mathcal{A}:\Omega\times \mathbb{R}^n\longrightarrow\mathbb{R}^n$, $\mathcal{B}:\Omega\times\mathbb{R}\times \mathbb{R}^n\longrightarrow\mathbb{R}^n$, $\mathcal{C}:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ are Carat'eodory functions that satisfy some conditions and the right-hand side term $f-{\rm{div}}F$ belongs to $L^{p'}(\Omega,\omega_{2}^{1-p'})+\prod\limits_{j=1}^{n}L^{p'}(\Omega,\omega_{1}^{1-p'})$. We will use the Browder-Minty Theorem and the weighted Sobolev spaces theory to prove the existence and uniqueness of weak solution in the weighted Sobolev space $W^{1,p}_ 0(\Omega,\omega_1,\omega_{2})$.

Keywords

[1] A. Abbassi, C. Allalou and A. Kassidi, Topological degree methods for a Neumann problem governed by nonlinear
elliptic equation, Moroccan J. of Pure and Appl. Anal., 6(2) (2020) 231–242.
[2] A. Abbassi, C. Allalou and A. Kassidi, Existence results for some nonlinear elliptic equations via topological degree methods, J Elliptic Parabol Equ., 7(1) (2021) 121–136.
[3] A. Abbassi, C. Allalou and A. Kassidi, Existence of weak solutions for nonlinear p-elliptic problem by topological degree, Nonlinear Dyn. Syst. Theory., 20(3) (2020) 229–241.
[4] F. Behboudi and A. Razani, Two weak solutions for a singular (p, q)−Laplacian problem, Filomat, 33(11) (2019) 3399–3407.
[5] A. Bensoussan, L. Boccardo and F. Murat, On a non linear partial differential equation having natural growth terms and unbounded solution, Annales de l’Institut Henri Poincare (C) Non Linear Anal., 5(4) (1988) 347–364.
[6] D. Bresch, J. Lemoine, F. Guillen-Gonzalez, A note on a degenerate elliptic equations with applications to lake and seas, Electron. J. Differ. Equations, 2004(42) (2004) 1–13.
[7] F. Chiarenza, Regularity for Solutions of Quasilinear Elliptic Equations Under Minimal Assumptions, Potential Theory Degenerate Partial Differ. Operators. Springer, Dordrecht (1995) 325–334
[8] M. Chipot, Elliptic Equations: An Introductory Course, Birkh¨auser, Berlin, 2009.
[9] 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, Pisa. 22 (2017).
[10] P. Drabek, A. Kufner, V. Mustonen, Pseudo-monotonicity and degenerated or singular elliptic operators, Bull.
Aust. Math. Soc., 58(2) (1998) 213–221.
[11] P. Drabek, A. Kufner and F. Nicolosi. Quasilinear elliptic equations with degenerations and singularities, Walter de Gruyter. 5(2011).
[12] E. Fabes, D. S. Jerison and C. E. Kenig, The Wiener test for degenerate elliptic equations, Annales de l’institut Fourier. 32(3) (1982) 151–182.
[13] E. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Commun. Stat.- Theory Methods, 7(1) (1982) 77–116.
[14] B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. 14(4) (1987) 527–568.
[15] S. Fucik, O. John and A. Kufner, Function Spaces, Noordhoff International Publishing, Leyden. Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977.
[16] J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, Elsevier, 2011.
[17] S. Heidari and A. Razani, Infinitely many solutions for (p(x), q(x))−Laplacian-like systems, Commun. Korean Math. Soc., 36(1) (2021) 51—62.
[18] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math, Monographs, Clarendon Press, Oxford, 1993.
[19] A. Kufner, Weighted sobolev spaces, John Wiley and Sons Incorporated 31 (1985).
[20] A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces , Commentationes Mathematicae Universitatis Carolinae., 25(3) (1984) 537–554.
[21] M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J., (2020) 1–10.
[22] B. Muckenhoupt, Weighted norm inequalities for the hardy maximal function, Trans. Am. Math. Soc., 165 (1972) 207–226.
[23] B. Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math., 49(1) (1974) 101–106.
[24] M. E. Ouaarabi, A. Abbassi and C. Allalou, Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces, J. Elliptic. Parabol. Equ., 7(1) (2021) 221–242.
[25] M. E. Ouaarabi, C. Allalou and A. Abbassi, On the Dirichlet Problem for some Nonlinear Degenerated Elliptic Equations with Weight, 2021 7th International Conference on Optimization and Applications (ICOA), 2021, pp. 1–6. doi: 10.1109/ICOA51614.2021.9442620.
[26] M. E. Ouaarabi, A. Abbassi and C. Allalou, Existence Result for a General Nonlinear Degenerate Elliptic Problems with Measure Datum in Weighted Sobolev Spaces, Int. J. Optim. Appl. , 1(2) (2021) 1–9.
[27] F. Safari and A. Razani, Positive weak solutions of a generalized supercritical Neumann problem, Iranian Journal
of Science and Technology, Trans. A: Sci.. 44(6) (2020) 1891–1898.
[28] A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, 1986.
[29] B.O.Turesson, Nonlinear potential theory and weighted Sobolev spaces, Springer Science and Business Media. 1736
(2000).[30] X. Xu, A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to
the thermistor problem, Differ. Integral Equ. 12(1) (1999) 83–100.
[31] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol.I, Springer-Verlag, Berlin, 1990.
[32] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol.II/B, Springer-Verlag, New York, 1990.
Volume 13, Issue 1
March 2022
Pages 2635-2653
  • Receive Date: 05 June 2021
  • Revise Date: 24 August 2021
  • Accept Date: 30 October 2021