Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator
DOI:
https://doi.org/10.5269/bspm.62772Resumo
Using the variational method, we prove the
existence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.
Referências
1. A. Boukhsas, B. Ouhamou, Steklov eigenvalues problems for generalized (p, q)-Laplacian type operators. To appear.
2. D.G. Costa, An Invitation to Variational Methods in Differential Equations. Birkhauser, Basel (2007)
3. P. Clement, R. Manasevich and E. Mitidieri, On a modified capillary equation, Journal of Differential Equations 124 (1996), 343-358.
4. D.E. Edmunds, J. Rakosnık, Sobolev embedding with variable exponent. Stud. Math. 143, 267-293 (2000).
5. D.E. Edmunds, J. Lang, A. Nekvinda, On Lp(x)(Ω) norms. Proc. R. Soc. Ser. A. 455, 219-225 (1999)
6. X.L. Fan, Regularity of minimizers of variational integrals with p(x)-growth conditions, Ann. Math. Sinica, 17 A(5) (1996), 557-564.
7. X.L. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω) , J. Math. Anal. Appl. 263 (2001), 424-446.
8. X.L. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces W1,p(x). J. Math. Anal. Appl. 262(2), 749.760 (2001)
9. X.L. Fan and D. Zhao, On the generalized Orlicz-Sobolev space Wk,p(x)(Ω). J.Gancu Educ. College 12 (1998), no. 1, 1-6.
10. Y. Jabri, The mountain Pass Theorem. Variants, Generalizations and Some Application, Cambridge University Press, (2003).
11. B. Karim, A. Zerouali and O. Chakrone, Existence and Multiplicity Results for Steklov Problems with p (.)-Growth Conditions. Bulletin of the Iranian Mathematical Society, 44(3), 819-836.(2018)
12. O. Kovacik,, J. Rakosnk, On spaces Lp(x)(Ω) and Wk,p(x)(Ω). Czechoslov. Math. J. 41(4), 592-618 (1991)
13. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
14. M. Mihailescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425.
15. W.M. Ni, J. Serrin, Existence and non-existence theorem s for ground states for quasilinear partial differential equations, Att. Conveg. Lincei 77 (1985), 231-257.
16. W.M. Ni, J. Serrin, Nonexistence theorems for quasilinear partial differential equations. Rend. Circ. Mat. Palermo (2) Suppl, 8, 171-185.(1985)
17. F. Obersnel, P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation. J. Differ. Equ. 249, 1674-1725 (2010).
18. M.E. Ouaarabi, C. Allalou, and S . Melliani, p (x)-Laplacian-Like Neumann Problems in Variable-Exponent Sobolev Spaces Via Topological Degree Methods. arXiv preprint arXiv:2112.06262.(2021) .
19. M.M. Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators. Mediterr. J. Math. 9, 211-223 (2012).
20. M. Willem, Minimax Theorems, Birkhauser, Basel, 1996.
2. D.G. Costa, An Invitation to Variational Methods in Differential Equations. Birkhauser, Basel (2007)
3. P. Clement, R. Manasevich and E. Mitidieri, On a modified capillary equation, Journal of Differential Equations 124 (1996), 343-358.
4. D.E. Edmunds, J. Rakosnık, Sobolev embedding with variable exponent. Stud. Math. 143, 267-293 (2000).
5. D.E. Edmunds, J. Lang, A. Nekvinda, On Lp(x)(Ω) norms. Proc. R. Soc. Ser. A. 455, 219-225 (1999)
6. X.L. Fan, Regularity of minimizers of variational integrals with p(x)-growth conditions, Ann. Math. Sinica, 17 A(5) (1996), 557-564.
7. X.L. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω) , J. Math. Anal. Appl. 263 (2001), 424-446.
8. X.L. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces W1,p(x). J. Math. Anal. Appl. 262(2), 749.760 (2001)
9. X.L. Fan and D. Zhao, On the generalized Orlicz-Sobolev space Wk,p(x)(Ω). J.Gancu Educ. College 12 (1998), no. 1, 1-6.
10. Y. Jabri, The mountain Pass Theorem. Variants, Generalizations and Some Application, Cambridge University Press, (2003).
11. B. Karim, A. Zerouali and O. Chakrone, Existence and Multiplicity Results for Steklov Problems with p (.)-Growth Conditions. Bulletin of the Iranian Mathematical Society, 44(3), 819-836.(2018)
12. O. Kovacik,, J. Rakosnk, On spaces Lp(x)(Ω) and Wk,p(x)(Ω). Czechoslov. Math. J. 41(4), 592-618 (1991)
13. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
14. M. Mihailescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425.
15. W.M. Ni, J. Serrin, Existence and non-existence theorem s for ground states for quasilinear partial differential equations, Att. Conveg. Lincei 77 (1985), 231-257.
16. W.M. Ni, J. Serrin, Nonexistence theorems for quasilinear partial differential equations. Rend. Circ. Mat. Palermo (2) Suppl, 8, 171-185.(1985)
17. F. Obersnel, P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation. J. Differ. Equ. 249, 1674-1725 (2010).
18. M.E. Ouaarabi, C. Allalou, and S . Melliani, p (x)-Laplacian-Like Neumann Problems in Variable-Exponent Sobolev Spaces Via Topological Degree Methods. arXiv preprint arXiv:2112.06262.(2021) .
19. M.M. Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators. Mediterr. J. Math. 9, 211-223 (2012).
20. M. Willem, Minimax Theorems, Birkhauser, Basel, 1996.
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2024-05-08
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