Solutions for nonhomogeneous degenerate quasilinear anisotropic problems
Year 2024,
, 134 - 149, 15.09.2024
Abdolrahman Razani
,
Elisabetta Tornatore
Abstract
In this article, we consider a class of nonlinear elliptic problems, where anisotropic leading differential operator incorporates the unbounded coefficients and the nonlinear term is a convection term. We prove the solvability of degenerate Dirichlet problem with convection, i.e. the existence of at least one bounded weak solution via the theory of pseudomonotone operators, Nemytskii-type operator and a priori estimate in the degenerate anisotropic Sobolev spaces.
References
- Y. Ahakkoud, J. Bennouna and M. Elmassoudi: Existence of a renormalized solutions for parabolic-elliptic system in anisotropic Orlicz-Sobolev spaces, Rend. Circ. Mat. Palermo, II. Ser (2024).
- M. Allalou, M. El Ouaarabi and A. Raji: On a class of nonhomogeneous anisotropic elliptic problem with variable exponents, Rend. Circ. Mat. Palermo, II. Ser (2024).
- M. Bohner, G. Caristi, A. Ghobadi and S. Heidarkhani: Three solutions for discrete anisotropic Kirchhoff-type
problems, Dem. Math., 56 (1) (2023), Article ID: 20220209.
- B. Brandolini, F. C. Cîrstea: Singular anisotropic elliptic equations with gradient-dependent lower order terms, Nonlinear Differ. Equ. Appl. NoDEA, 30 (2023), Article ID:58.
- B. Brandolini, F.C. Cîrstea: Anisotropic elliptic equations with gradient-dependent lower order terms in L1 data, Mathematics in Engineering, 5 (4) (2023), 1–33.
- S. Carl, V. K. Le and D. Motreanu: Nonsmooth variational problems and their inequalities, in: Comparison Principles and Applications, Springer, New York (2007).
- S. Ciani, V. Vespri: On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation, Constr. Math. Anal., 4 (1) (2021), 93–103.
- G. di Blasio, F. Feo and G. Zecca: Regularity results for local solutions to some anisotropic elliptic equations, Isr. J. Math., 261 (2023), 1–35.
- G. di Blasio, F. Feo and G. Zecca: Existence and uniqueness of solutions to some anisotropic elliptic equations with singular convection term, https://doi.org/10.48550/arXiv.2307.13564
- P. Drabek, A. Kufner and F. Nicolosi: Quasilinear Eliptic Equations with Degenerations and Singularities, De Gruyter Series in Nonlinear Analysis and Applications, 5; Walter de Gruyter & Co.: Berlin, Germany (1997).
- X. Fan: Anisotropic variable exponent Sobolev spaces and −→p (x)-Laplacian equations, Complex Var. Elliptic Equ., 56 (79) (2011), 623–642.
- I. Fragala, F. Gazzola and B. Kawohl: Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (5) (2004), 715–734.
- V. Gutlyanskii, O. Nesmelova, V. Ryazanov and E. Yakubov: Toward the theory of semi-linear Beltrami equations, Constr. Math. Anal., 6 (3) (2023), 151–163.
- M. Mih˘ailescu, G. Moro¸sanu: On an eigenvalue problem for an anisotropic elliptic equation involving variable exponents, Glasgow Math. J., 52 (2010), 517–527.
- M. Mih˘ailescu, P. Pucci and V. D. R˘adulescu: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687–698.
- D. Motreanu: Degenerated and competing dirichlet problems with weights and convection, Axioms, 10 (4) (2021), Article ID: 271.
- D. Motreanu, E. Tornatore: Quasilinear Dirichlet problems with degenerated p-Laplacian and convection term, Mathematics, 9 (2) (2021), Article ID: 139.
- D. Motreanu, E. Tornatore: Nonhomogeneous degenerate quasilinear problems with convection, Nonlinear Anal. Real World Appl., 71 (2023), Article ID: 103800.
- D. Motreanu, E. Tornatore: Dirichlet problems with anisotropic principal part involving unbounded coefficients, Electron. J. Differ. Equ., 2024 (11), 1–13.
- V.A. Nghiem Thi, A.T. Vu, D.L. Le and V.N. Doan: On the source problem for the diffusion equations with conformable derivative, Modern Math. Methods, 2 (2) (2024), 55–64.
- V. D. R˘adulescu: Isotropic and anisotropic double-phase problems: Old and new, Opuscula Math., 39 (2) (2019), 259–279.
- J. Rákosník: Some remarks to anisotropic Sobolev spaces I, Beitr. Anal., 13 (1979), 55–68.
- A. Razani: Nonstandard competing anisotropic (p, q)-Laplacians with convolution, Bound. Value Probl., 2022 (2022), Article ID:87.
- A. Razani: Entire weak solutions for an anisotropic equation in the Heisenberg group, Proc. Amer. Math. Soc., 151 (11) (2023), 4771–4779.
- A. Razani, G.S. Costa and G. M. Figueiredo: A positive solution for a weighted anisotropic p-Laplace equation involving vanishing potential, Mediterr. J. Math., 21 (2024), Article ID: 59.
- A. Razani, G. M. Figueiredo: A positive solution for an anisotropic p&q-Laplacian, Discrete Contin. Dyn. Syst. Ser. S, 16 (6) (2023), 1629–1643.
- A. Razani, G. M. Figueiredo: Existence of infinitely many solutions for an anisotropic equation using genus theory, Math. Methods Appl. Sci., 45 (12) (2022), 7591–7606.
- A. Razani, G. M. Figueiredo: Degenerated and competing anisotropic (p, q)-Laplacians with weights, Appl. Anal., 102 (16) (2023), 4471–4488.
Year 2024,
, 134 - 149, 15.09.2024
Abdolrahman Razani
,
Elisabetta Tornatore
References
- Y. Ahakkoud, J. Bennouna and M. Elmassoudi: Existence of a renormalized solutions for parabolic-elliptic system in anisotropic Orlicz-Sobolev spaces, Rend. Circ. Mat. Palermo, II. Ser (2024).
- M. Allalou, M. El Ouaarabi and A. Raji: On a class of nonhomogeneous anisotropic elliptic problem with variable exponents, Rend. Circ. Mat. Palermo, II. Ser (2024).
- M. Bohner, G. Caristi, A. Ghobadi and S. Heidarkhani: Three solutions for discrete anisotropic Kirchhoff-type
problems, Dem. Math., 56 (1) (2023), Article ID: 20220209.
- B. Brandolini, F. C. Cîrstea: Singular anisotropic elliptic equations with gradient-dependent lower order terms, Nonlinear Differ. Equ. Appl. NoDEA, 30 (2023), Article ID:58.
- B. Brandolini, F.C. Cîrstea: Anisotropic elliptic equations with gradient-dependent lower order terms in L1 data, Mathematics in Engineering, 5 (4) (2023), 1–33.
- S. Carl, V. K. Le and D. Motreanu: Nonsmooth variational problems and their inequalities, in: Comparison Principles and Applications, Springer, New York (2007).
- S. Ciani, V. Vespri: On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation, Constr. Math. Anal., 4 (1) (2021), 93–103.
- G. di Blasio, F. Feo and G. Zecca: Regularity results for local solutions to some anisotropic elliptic equations, Isr. J. Math., 261 (2023), 1–35.
- G. di Blasio, F. Feo and G. Zecca: Existence and uniqueness of solutions to some anisotropic elliptic equations with singular convection term, https://doi.org/10.48550/arXiv.2307.13564
- P. Drabek, A. Kufner and F. Nicolosi: Quasilinear Eliptic Equations with Degenerations and Singularities, De Gruyter Series in Nonlinear Analysis and Applications, 5; Walter de Gruyter & Co.: Berlin, Germany (1997).
- X. Fan: Anisotropic variable exponent Sobolev spaces and −→p (x)-Laplacian equations, Complex Var. Elliptic Equ., 56 (79) (2011), 623–642.
- I. Fragala, F. Gazzola and B. Kawohl: Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (5) (2004), 715–734.
- V. Gutlyanskii, O. Nesmelova, V. Ryazanov and E. Yakubov: Toward the theory of semi-linear Beltrami equations, Constr. Math. Anal., 6 (3) (2023), 151–163.
- M. Mih˘ailescu, G. Moro¸sanu: On an eigenvalue problem for an anisotropic elliptic equation involving variable exponents, Glasgow Math. J., 52 (2010), 517–527.
- M. Mih˘ailescu, P. Pucci and V. D. R˘adulescu: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687–698.
- D. Motreanu: Degenerated and competing dirichlet problems with weights and convection, Axioms, 10 (4) (2021), Article ID: 271.
- D. Motreanu, E. Tornatore: Quasilinear Dirichlet problems with degenerated p-Laplacian and convection term, Mathematics, 9 (2) (2021), Article ID: 139.
- D. Motreanu, E. Tornatore: Nonhomogeneous degenerate quasilinear problems with convection, Nonlinear Anal. Real World Appl., 71 (2023), Article ID: 103800.
- D. Motreanu, E. Tornatore: Dirichlet problems with anisotropic principal part involving unbounded coefficients, Electron. J. Differ. Equ., 2024 (11), 1–13.
- V.A. Nghiem Thi, A.T. Vu, D.L. Le and V.N. Doan: On the source problem for the diffusion equations with conformable derivative, Modern Math. Methods, 2 (2) (2024), 55–64.
- V. D. R˘adulescu: Isotropic and anisotropic double-phase problems: Old and new, Opuscula Math., 39 (2) (2019), 259–279.
- J. Rákosník: Some remarks to anisotropic Sobolev spaces I, Beitr. Anal., 13 (1979), 55–68.
- A. Razani: Nonstandard competing anisotropic (p, q)-Laplacians with convolution, Bound. Value Probl., 2022 (2022), Article ID:87.
- A. Razani: Entire weak solutions for an anisotropic equation in the Heisenberg group, Proc. Amer. Math. Soc., 151 (11) (2023), 4771–4779.
- A. Razani, G.S. Costa and G. M. Figueiredo: A positive solution for a weighted anisotropic p-Laplace equation involving vanishing potential, Mediterr. J. Math., 21 (2024), Article ID: 59.
- A. Razani, G. M. Figueiredo: A positive solution for an anisotropic p&q-Laplacian, Discrete Contin. Dyn. Syst. Ser. S, 16 (6) (2023), 1629–1643.
- A. Razani, G. M. Figueiredo: Existence of infinitely many solutions for an anisotropic equation using genus theory, Math. Methods Appl. Sci., 45 (12) (2022), 7591–7606.
- A. Razani, G. M. Figueiredo: Degenerated and competing anisotropic (p, q)-Laplacians with weights, Appl. Anal., 102 (16) (2023), 4471–4488.