Abstract
Based on the recent Berkovits degree, and by way of an abstract Hammerstein equation, we study the Dirichlet boundary value problem involving nonlinear operators of the form
where a and f are Carathéodory functions satisfying some nonstandard growth conditions and \(\lambda \in I\!R\) . The function a satisfy also a condition of strict monotony and a condition of coercivity. We prove the existence of weak solutions of this problem in the weighted Sobolev spaces \(W_0^{1,p(x)}(\varOmega ,\rho )\) where \(\rho \) is a weight function, satisfying some integrability conditions.
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References
Aydin, I.: Weighted variable sobolev spaces and capacity. J. Funct. Spaces Appl. 2012, 17 pages (2012). Article ID 132690
Azroul, E., Benboubker, M.B., Barbara, A.: Quasilinear elliptic problems with nonstandard growth. EJDE 2011(62), 1–16 (2011)
Berkovits, J.: On the degree theory for nonlinear mappings of monotone type. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 58, 2 (1986)
Berkovits, J.: Extension of the Leray-Schauder degree for abstract Hammerstein type mappings. J. Differ. Equ. 234, 289–310 (2007)
Berkovits, J., Mustonen, V.: On topological degree for mappings of monotone type. Nonlinear Anal. 10, 1373–1383 (1986)
Berkovits, J., Mustonen, V.: Nonlinear mappings of monotone type I. Classification and degree theory, Preprint No 2/88, Mathematics, University of Oulu
Boureanu, M.M., Matei, A., Sofonea, M.: Nonlinear problems with \(p(x)\)-growth conditions and applications to antiplane contact models. Adv. Nonlinear Stud. 14, 295–313 (2014)
Diening, L., et al.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-18363-8
Fan, X.L., Zhang, Q.H.: Existence for \(p(x)\)-Laplacien Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)
Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{k, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)
Kovàčik, O., Rákosnik, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czechoslovak Math. J. 41, 592–618 (1991)
Lahmi, B., Azroul, E., El Haiti, K.: Nonlinear degenerated elliptic problems with dual data and nonstandard growth. Math. Rep. 20(70), 81–91 (2018)
Rajagopal, K.R., Ru̇žička, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)
Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)
Samko, S.G.: Density of \({\cal C}_0^\infty ({I\!R}^N)\) in the generalized Sobolev spaces \(W^{m, p(x)}({I\!R}^N)\). Dokl. Akad. Nauk. 369(4), 451–454 (1999)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators. Springer, New York (1990)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory, [in Russian]. Izv. Akad. Nauk SSSR, Ser. Math. 50(4), 675–710 (1986). English translation: Math. USSR, Izv. 29(1), 33–66 (1987)
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Ait Hammou, M., Azroul, E. (2020). Nonlinear Elliptic Boundary Value Problems by Topological Degree. In: Zerrik, E., Melliani, S., Castillo, O. (eds) Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Studies in Systems, Decision and Control, vol 243. Springer, Cham. https://doi.org/10.1007/978-3-030-26149-8_1
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