Existence results for some nonlinear degenerate problems in the anisotropic spaces
DOI:
https://doi.org/10.5269/bspm.41366Resumo
Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:
- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,
where for $i=1,...,N$ $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and $H_i(x,u,\nabla u)$ is a nonlinear term without a sign condition. Under suitable conditions on $a_i$ and $H_i$, we prove the existence of weak solutions.
Referências
1. Lions, J. L., Exact Controllability, Stabilizability, and Perturbations for Distributed Systems, Siam Rev. 30, 1-68, (1988).
2. C. M. Dafermos, C. M., An abstract Volterra equation with application to linear viscoelasticity. J. Differential Equations 7, 554-589, (1970).
3. A. Alvino, M.F. Betta, and A. Mercaldo, Comparison principle for some classes of nonlinear elliptic equations. J. Differential Equations 249, 3279-3290, (2010).
4. S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results. Differential Integral Equations 21, 401-419, (2008).
5. Bendahmane M, Langlais M, Saad M, On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease. Nonlinear Analysis, 54(4):617-636, (2003).
6. M. F. Betta, A. Mercaldo F. Murat c, M. M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure. J. Math. Pures Appl. 82, 90-124, (2003).
7. Boccardo L, Gallouet T, Marcellini P, Anisotropic equations in L1. Differential and Integral Equations, 9: 209-212, (1996).
8. L. Boccardo, T. Gallouet, JL. Vasquez, Nonlinear elliptic equations in RN without restrictions on the data. Journal of Differential Equations, 105(2):334-363, (1993)
9. L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19, 581-597, (1992).
10. L. Boccardo, D. Giachetti, J. I. Diaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differential Equations 106, no. 2, 215-237, (1993).
11. M. Bojowald, H H. Hernandez, H A. Morales Tecotl, Perturbative degrees of freedom in loop quantum gravity: anisotropies. Classical Quantum Gravity 10: 3491-3516, (2006).
12. G. Bottaro and M. Marina, Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati, Boll. Un. Mat. Ital. 8 , 46-56, (1973).
13. H. Bresis, FE. Browder, Some properties of higher order Sobolev spaces, J. Math pures et appli. 61, 245-259, (1982).
14. A. Dall’Aglio, Approximated solutions of equations with L1 data. Application to the Hconvergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. (4) 170, 207-240, (1996).
15. T. Del Vecchio and M. M. Porzio , Existence results for a class of non- coercive Dirichlet problems, Ricerche Mat. 44 (1995), 421-438 (1996).
16. R. Di Nardo , F. Feo, Existence and uniqueness for nonlinear anisotropic elliptic equations. Arch. Math. 102, 141-153, (2014).
17. R. Di Nardo , F. Feo , and O. Guide , Uniqueness result for nonlinear anisotropic elliptic equations. Adv. Differential Equations 18, 433–458, (2013).
18. I. Fragala, F. Gazzola, and B. Kawohl , Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. H. Poincare Anal. Non Linaire 21, 715-734, (2004)
19. J. L. Lions; Quelques methodes de resolution des problemes aux limites non lineaires. Dunod et Gauthier-Villars, Paris, (1969).
20. F. Murat. Solutions renormalizadas de edp elipticas no lineales. Publ. Laboratoire d’Analyse Numrique, Univ. Paris 6, R 93023, (1993).
21. A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. (4) 17, 143-172,(1999).
22. J. M. Rakotoson, Existence of bounded solutions of some degenerate quasilinear elliptic equations, Comm. Partial Differential Equations 12 (6), 633-676, (1987).
23. M. Troisi , Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 18 , 3-24, (1969).
24. J. Vetois, A priori estimates for solutions of anisotropic elliptic equations. Nonlinear Analysis 71(9): 3881-3905, (2009).
25. W. Zou, F. Li, Existence of solutions for degenerate quasilinear elliptic equations. J.Nonlinear Analysis 73 3069-3082, (2010)
2. C. M. Dafermos, C. M., An abstract Volterra equation with application to linear viscoelasticity. J. Differential Equations 7, 554-589, (1970).
3. A. Alvino, M.F. Betta, and A. Mercaldo, Comparison principle for some classes of nonlinear elliptic equations. J. Differential Equations 249, 3279-3290, (2010).
4. S. Antontsev and M. Chipot, Anisotropic equations: uniqueness and existence results. Differential Integral Equations 21, 401-419, (2008).
5. Bendahmane M, Langlais M, Saad M, On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease. Nonlinear Analysis, 54(4):617-636, (2003).
6. M. F. Betta, A. Mercaldo F. Murat c, M. M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure. J. Math. Pures Appl. 82, 90-124, (2003).
7. Boccardo L, Gallouet T, Marcellini P, Anisotropic equations in L1. Differential and Integral Equations, 9: 209-212, (1996).
8. L. Boccardo, T. Gallouet, JL. Vasquez, Nonlinear elliptic equations in RN without restrictions on the data. Journal of Differential Equations, 105(2):334-363, (1993)
9. L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19, 581-597, (1992).
10. L. Boccardo, D. Giachetti, J. I. Diaz and F. Murat, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differential Equations 106, no. 2, 215-237, (1993).
11. M. Bojowald, H H. Hernandez, H A. Morales Tecotl, Perturbative degrees of freedom in loop quantum gravity: anisotropies. Classical Quantum Gravity 10: 3491-3516, (2006).
12. G. Bottaro and M. Marina, Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati, Boll. Un. Mat. Ital. 8 , 46-56, (1973).
13. H. Bresis, FE. Browder, Some properties of higher order Sobolev spaces, J. Math pures et appli. 61, 245-259, (1982).
14. A. Dall’Aglio, Approximated solutions of equations with L1 data. Application to the Hconvergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. (4) 170, 207-240, (1996).
15. T. Del Vecchio and M. M. Porzio , Existence results for a class of non- coercive Dirichlet problems, Ricerche Mat. 44 (1995), 421-438 (1996).
16. R. Di Nardo , F. Feo, Existence and uniqueness for nonlinear anisotropic elliptic equations. Arch. Math. 102, 141-153, (2014).
17. R. Di Nardo , F. Feo , and O. Guide , Uniqueness result for nonlinear anisotropic elliptic equations. Adv. Differential Equations 18, 433–458, (2013).
18. I. Fragala, F. Gazzola, and B. Kawohl , Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. H. Poincare Anal. Non Linaire 21, 715-734, (2004)
19. J. L. Lions; Quelques methodes de resolution des problemes aux limites non lineaires. Dunod et Gauthier-Villars, Paris, (1969).
20. F. Murat. Solutions renormalizadas de edp elipticas no lineales. Publ. Laboratoire d’Analyse Numrique, Univ. Paris 6, R 93023, (1993).
21. A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. (4) 17, 143-172,(1999).
22. J. M. Rakotoson, Existence of bounded solutions of some degenerate quasilinear elliptic equations, Comm. Partial Differential Equations 12 (6), 633-676, (1987).
23. M. Troisi , Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 18 , 3-24, (1969).
24. J. Vetois, A priori estimates for solutions of anisotropic elliptic equations. Nonlinear Analysis 71(9): 3881-3905, (2009).
25. W. Zou, F. Li, Existence of solutions for degenerate quasilinear elliptic equations. J.Nonlinear Analysis 73 3069-3082, (2010)
Downloads
Publicado
2020-10-11
Edição
Seção
Artigos
Licença
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
