International Journal of Nonlinear Analysis and Applications
Existence results of some p(u)-Laplacian systems
Document Type : Research Paper
Authors
LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco
Abstract
In this paper, we consider the existence of weak solutions for some $p(u)$-Laplacian problems with Dirichlet boundary conditions. Here the exponent of nonlinearity $p$ depends on the solution $u$ itself. Existence results for the associated boundary-value local problem are given by using a singular perturbation technique combined with the theory of Sobolev spaces with exponent variables.
Keywords
[1] S. Antontsev, S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions. Existence, Uniqueness, Localization, Blow-up, Atlantis Press, Paris, 2015.
[2] S.N. Antontsev, J.F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat. 52 (2006), no 1, 19–36.
[3] B. Andreianov, M. Bendahmane, S. Ouaro, Structural stability for variable exponent elliptic problems. II. The p(u)-Laplacian and coupled problems, Nonlinear Anal. 72 (2010), no. 12, 4649–4660.
[4] P. Blomgren, T. Chan, P. Mulet and C. Wong, Total variation image restoration: Numerical methods and extensions, Proc. IEEE Int. Conf. Image Process. vol. 3. IEEE Comput. Soc. Press, Piscataway, 1997, pp. 384–387.
[5] E. Bollt, R. Chartrand, S. Esedoglu, P. Schultz and K. Vixie, Graduated, adaptive image denoising: local compromise between total-variation and isotropic diffusion, Adv. Comput. Math. 31 (2007), 61–85.
[6] M. Chipot and H. B. de Oliveira, Some Results On The p(u)-Laplacian Problem, Math. Annal. 375 (2019), no. 1, 283–306.
[7] M. Chipot , Elliptic Equations: An Introductory Course, Birkh¨auser, Basel, 2009.
[8] M. Chipot and J.F. Rodrigues, On a class of non-linear nonlocal elliptic problems, M2AN 26 (1992), no. 3, 447-–468.
[9] M. Chipot, B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Dyn. Contin. Discr. Impuls. Syst. Ser. A Math. Anal. 8 (2001), no. 1, 35-–51.
[9] M. Chipot, B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Dyn. Contin. Discr. Impuls. Syst. Ser. A Math. Anal. 8 (2001), no. 1, 35-–51.
[10] D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Birkh¨auser/Springer, Heidelberg, 2013.
[11] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383—1406.
[12] L. Diening, P. Harjulehto, P. H¨ast¨o and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer, 2011.
[13] R. Glowinski, R. Marrocco, Sur l’approximation, par elements finis d’ordre un, et la resolution, par penalisationdualit´e, d’une classe de probl`emes de Dirichlet non lin´eaires, RAIRO Ana. Numer. 9 (1975), 41–76.
[14] B. Lovat, Etude de quelques probl`emes paraboliques non locaux, Th`ese, Universit´e de Metz, 1995.
[15] A.L. Skubachevskii, Elliptic functional differential equations and applications, Birkh¨auser, 1997.
[16] K. Rajagopal and M. Ruzicka, Mathematical modelling of electro-rheological fluids, Contin. Mech. Thermodyn. 13 (2001), 59—78.
[17] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer, Berlin, 2000.
[18] J. T¨urola, Image denoising using directional adaptive variable exponents model, J. Math. Imaging. Vis. 57 (2017), 56–74.
[19] V.V. Zhikov, On the density of smooth functions in Sobolev–Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226 (2004), 67-–81.
[20] V.V. Zhikov, On some variational problems, Russ. J. Math. Phys. 5 (1997-1998), no. 1, 105–116.
[21] V.V. Zhikov, Meyers-type estimates for solving the nonlinear Stokes system, Differ. Uravn. 33 (1997), no. 1,
107–114 (in Russian); English Transl. Differ. Equ. 33 (1997), no. 1, 108–115.
[22] V.V. Zhikov, Solvability of the three-dimensional termistor problem, Trudy Mat. Inst. Steklov 261 (2008) 101–114
(in Russian); Engl. transl. in Proc. Steklov Inst. Math. 261 (2008), 1–14.
[23] V.V. Zhikov, On passage to the limit in nonlinear elliptic equations, Dokl. Math. 77 (2008), no. 3, 383–387.
[21] V.V. Zhikov, Meyers-type estimates for solving the nonlinear Stokes system, Differ. Uravn. 33 (1997), no. 1,
107–114 (in Russian); English Transl. Differ. Equ. 33 (1997), no. 1, 108–115.
[22] V.V. Zhikov, Solvability of the three-dimensional termistor problem, Trudy Mat. Inst. Steklov 261 (2008) 101–114
(in Russian); Engl. transl. in Proc. Steklov Inst. Math. 261 (2008), 1–14.
[23] V.V. Zhikov, On passage to the limit in nonlinear elliptic equations, Dokl. Math. 77 (2008), no. 3, 383–387.
)
Volume 13, Issue 2
July 2022Pages 3073-3082
- Receive Date: 10 May 2022
- Accept Date: 04 July 2022