Existence results in weighted Sobolev space for quasilinear degenerate p(z)−elliptic problems with a Hardy potential
Abstract
The objective of this work is to establish the existence of entropy solutions to degenerate nonlinear elliptic problems for L1-data f with a Hardy potential, in weighted Sobolev spaces with variable exponent, which are represented as follows 
where 
is a Leray-Lions operator from 
into its dual, 
is a non-linearity term that only meets the growth condition, and ρ > 0 is a constant.
Keyword : nonlinear elliptic equations, entropy solutions, Hardy potential, weighted variable exponent Sobolev space
How to Cite
Zineddaine, G., Sabiry, A., Melliani, S., & Kassidi, A. (2024). Existence results in weighted Sobolev space for quasilinear degenerate p(z)−elliptic problems with a Hardy potential. Mathematical Modelling and Analysis, 29(3), 460–479. https://doi.org/10.3846/mma.2024.18696
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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E. Azroul, A. Barbara and H. Hjiaj. Strongly nonlinear p(x)-elliptic problems with L1-data. African Diaspora Journal of Mathematics, 16(2):1–22, 2014.
E. Azroul, M.B. Benboubker, H. Hjiaj and C. Yazough. Existence of solutions for a class of obstacle problems with L1-data and without sign condition. Afrika Matematika, 27:795–813, 2016. https://doi.org/10.1007/s13370-015-0375-y
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O. Benslimane, A. Aberqi and J. Bennouna. Existence results for double phase obstacle problems with variable exponents. Journal of Elliptic and Parabolic Equations, 7:875–890, 2021. https://doi.org/10.1007/s41808-021-00122-z
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D. Bresch, J. Lemoine and F. Guillen-Gonzalez. A note on a degenerate elliptic equation with applications for lakes and seas. Electronic Journal of Differential Equations (EJDE), 2004:13, 2004.
H. Brézis and W. Strauss. Semi-linear second-order elliptic equations in L1. Journal of the Mathematical Society of Japan, 25(4):565–590, 1973. https://doi.org/10.2969/jmsj/02540565
A.C. Cavalheiro. L∞-solutions for some degenerate quasilinear elliptic equations. Electronic Journal of Mathematical Analysis and Applications, 2(1):149– 163, 2004.
M. Colombo. Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations: With Applications to the Vlasov-Poisson and Semigeostrophic Systems, volume 22. Springer, 2017. https://doi.org/10.1007/978-88-7642-607-0
P. Drábek, A. Kufner and F. Nicolosi. Quasilinear elliptic equations with degenerations and singularities, volume 5. De Gruyter, 2011.
G. Di Fazio, D.K. Palagachev and M.A. Ragusa. Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. Journal of Functional Analysis, 166(2):179–196, 1999. https://doi.org/10.1006/jfan.1999.3425
A. Kalybay. Boundedness of one class of integral operators from second order weighted Sobolev space to weighted Lebesgue space. Journal of Function Spaces, 2022:Article 5257476. https://doi.org/10.1155/2022/5257476
J.L. Lions. Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Dunod, 1969.
V.M. Monetti and L. Randazzo. Existence results for nonlinear elliptic equations with p-growth in the gradient. Ricerche di Matematica, 49(1):163, 2000.
A. Porretta. Nonlinear equations with natural growth terms and measure data. Electronic Journal of Differential Equations (EJDE), 2002:183–202, 2002.
M.A. Ragusa. Linear growth coefficients in quasilinear equations. Nonlinear Differential Equations and Applications NoDEA, 13:605–617, 2007. https://doi.org/10.1007/s00030-006-4026-8
M.A. Ragusa and P. Zamboni. A potential theoretic inequality. Czechoslovak Mathematical Journal, 51:55–65, 2001. https://doi.org/10.1023/A:1013749603910
T. Del Vecchio. Nonlinear elliptic equations with measure data. Potential Analysis, 4:185–203, 1995. https://doi.org/10.1007/BF01275590
A. Youssfi, E. Azroul and H. Hjiaj. On nonlinear elliptic equations with hardy potential and L1-data. Monatshefte fu¨r Mathematik, 173:107–129, 2014. https://doi.org/10.1007/s00605-013-0516-z
A. Youssfi, A. Benkirane and M. El Moumni. Existence result for strongly nonlinear elliptic unilateral problems with L1data. Complex Variables and Elliptic Equations, 59(4):447–461, 2014. https://doi.org/10.1080/17476933.2012.725166
A. Abbassi, C. Allalou and A. Kassidi. Anisotropic elliptic nonlinear obstacle problem with weighted variable exponent. J. Math. Study, 54(4):337–356, 2021. https://doi.org/10.4208/jms.v54n4.21.01
A. Abbassi, C. Allalou and A. Kassidi. Existence of entropy solutions of the anisotropic elliptic nonlinear problem with measure data in weighted Sobolev space. Boletim da Sociedade Paranaense de Matema´tica, 40:1–22, 2022. https://doi.org/10.5269/bspm.52541
A. Abbassi, E. Azroul and E. Barbara. Degenerate p(x)-elliptic equation with second membre in L1. Adv. Sci. Technol. Eng. Syst. J, 2(5):45–54, 2017. https://doi.org/10.5269/bspm.52541
A. Aberqi, O. Benslimane, M. Elmassoudi and M.A. Ragusa. Nonnegative solution of a class of double phase problems with logarithmic nonlinearity. Boundary Value Problems,2022(1):57,2022. https://doi.org/10.1186/s13661-022-01639-5
Y. Akdim, E. Azroul and A. Benkirane. Existence of solutions for quasilinear degenerate elliptic equations. Electronic Journal of Differential Equations (EJDE), 2001:19, 2001.
E. Azroul, A. Barbara and H. Hjiaj. Strongly nonlinear p(x)-elliptic problems with L1-data. African Diaspora Journal of Mathematics, 16(2):1–22, 2014.
E. Azroul, M.B. Benboubker, H. Hjiaj and C. Yazough. Existence of solutions for a class of obstacle problems with L1-data and without sign condition. Afrika Matematika, 27:795–813, 2016. https://doi.org/10.1007/s13370-015-0375-y
E. Azroul, M. Bouziani and A. Barbara. Existence of entropy solutions for anisotropic quasilinear degenerated elliptic problems with Hardy potential. SeMA Journal,78:475–499,2021. https://doi.org/10.1007/s40324-021-00247-0
E. Azroul, H. Hjiaj and A. Touzani. Existence and regularity of entropy solutions for strongly nonlinear p(x)-elliptic equations. Electronic J. Diff. Equ, 68:1–27, 2013.
O. Benslimane, A. Aberqi and J. Bennouna. Existence results for double phase obstacle problems with variable exponents. Journal of Elliptic and Parabolic Equations, 7:875–890, 2021. https://doi.org/10.1007/s41808-021-00122-z
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 Poincaré C, Analyse non linéaire, 5(4):347–364, 1988. https://doi.org/10.1016/s0294-1449(16)30342-0
L. Boccardo and Th. Gallouët. Strongly nonlinear elliptic equations having natural growth terms and L1 data. Nonlinear Analysis: Theory, Methods & Applications, 19(6):573–579, 1992. https://doi.org/10.1016/0362-546X(92)90022-7
D. Bresch, J. Lemoine and F. Guillen-Gonzalez. A note on a degenerate elliptic equation with applications for lakes and seas. Electronic Journal of Differential Equations (EJDE), 2004:13, 2004.
H. Brézis and W. Strauss. Semi-linear second-order elliptic equations in L1. Journal of the Mathematical Society of Japan, 25(4):565–590, 1973. https://doi.org/10.2969/jmsj/02540565
A.C. Cavalheiro. L∞-solutions for some degenerate quasilinear elliptic equations. Electronic Journal of Mathematical Analysis and Applications, 2(1):149– 163, 2004.
M. Colombo. Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations: With Applications to the Vlasov-Poisson and Semigeostrophic Systems, volume 22. Springer, 2017. https://doi.org/10.1007/978-88-7642-607-0
P. Drábek, A. Kufner and F. Nicolosi. Quasilinear elliptic equations with degenerations and singularities, volume 5. De Gruyter, 2011.
G. Di Fazio, D.K. Palagachev and M.A. Ragusa. Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. Journal of Functional Analysis, 166(2):179–196, 1999. https://doi.org/10.1006/jfan.1999.3425
A. Kalybay. Boundedness of one class of integral operators from second order weighted Sobolev space to weighted Lebesgue space. Journal of Function Spaces, 2022:Article 5257476. https://doi.org/10.1155/2022/5257476
J.L. Lions. Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Dunod, 1969.
V.M. Monetti and L. Randazzo. Existence results for nonlinear elliptic equations with p-growth in the gradient. Ricerche di Matematica, 49(1):163, 2000.
A. Porretta. Nonlinear equations with natural growth terms and measure data. Electronic Journal of Differential Equations (EJDE), 2002:183–202, 2002.
M.A. Ragusa. Linear growth coefficients in quasilinear equations. Nonlinear Differential Equations and Applications NoDEA, 13:605–617, 2007. https://doi.org/10.1007/s00030-006-4026-8
M.A. Ragusa and P. Zamboni. A potential theoretic inequality. Czechoslovak Mathematical Journal, 51:55–65, 2001. https://doi.org/10.1023/A:1013749603910
T. Del Vecchio. Nonlinear elliptic equations with measure data. Potential Analysis, 4:185–203, 1995. https://doi.org/10.1007/BF01275590
A. Youssfi, E. Azroul and H. Hjiaj. On nonlinear elliptic equations with hardy potential and L1-data. Monatshefte fu¨r Mathematik, 173:107–129, 2014. https://doi.org/10.1007/s00605-013-0516-z
A. Youssfi, A. Benkirane and M. El Moumni. Existence result for strongly nonlinear elliptic unilateral problems with L1data. Complex Variables and Elliptic Equations, 59(4):447–461, 2014. https://doi.org/10.1080/17476933.2012.725166