Existence Results for a Dirichlet Boundary Value Problem Involving the \(p(x)\)-Laplacian Operator

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/s8393-0239-0126-b Existence Results for a Dirichlet Boundary Value Problem Involving the \(p(x)\)-Laplacian Operator Ait Hammou, M. Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 2. Abstract: The aim of this paper is to establish the existence of weak solutions, in \(W_0^{1,p(x)}(\Omega)\), for a Dirichlet boundary value problem involving the \(p(x)\)-Laplacian operator. Our technical approach is based on the Berkovits topological degree theory for a class of demicontinuous operators of generalized \((S_+)\) type. We also use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, and specially properties of \(p(x)\)-Laplacian operator. In order to use this theory, we will transform our problem into an abstract Hammerstein equation of the form \(v+S\circ Tv=0\) in the reflexive Banach space \(W^{-1,p'(x)}(\Omega)\) which is the dual space of \(W_0^{1,p(x)}(\Omega)\). Note also that the problem can be seen as a nonlinear eigenvalue problem of the form\(Au=\lambda u,\) where \(Au:=-Div(\|\nabla u\|^{p(x)-2}\nabla u)-f(x,u)\). When this problem admits a non-zero weak solution \(u\), \(\lambda\) is an eigenvalue of it and \(u\) is an associated eigenfunction. Keywords: Dirichlet problem, topological degree, \(p(x)\)-Laplacian operator. Language: English Download the full text For citation: Ait Hammou, M. Existence Results for a Dirichlet Boundary Value Problem Involving the \(p(x)\)-Laplacian Operator, Vladikavkaz Math. J., 2022, vol. 24, no. 2, pp. 5-13. DOI 10.46698/s8393-0239-0126-b + References 1.Brouwer, L. E. J. Uber Abbildung von Mannigfaltigkeiten, Mathematische Annalen, 1912, vol. 71, no. 1, pp. 97-115. DOI: 10.1007/BF01456931. 2. Leray, J. and Schauder, J. Topologie et Equationes Fonctionnelles, Annales Scientifiques de l'Ecole Normale Superieure, 1934, vol. 51, pp. 45-78. DOI: 10.24033/asens.836. 3. Berkovits, J. On the Degree Theory for Nonlinear Mappings of Monotone Type, Annales Academia Scientiarum Fennica. Series A I. Mathematica Dissertationes, 1986, vol. 58, 58 p. 4. Berkovits, J. Extension of the Leray-Schauder Degree for abstract Hammerstein Type Mappings, Journal of Differential Equations, 2007, vol. 234, no. 1, pp. 289-310. DOI: 10.1016/j.jde.2006.11.012. 5. Berkovits, J. and Mustonen, V. On Topological Degree for Mappings of Monotone Type, Nonlinear Analysis: Theory, Methods and Applications, 1986, vol. 10, no. 12, pp. 1373-1383. DOI: 10.1016/0362-546X(86)90108-2. 6. Berkovits, J. and Mustonen, V. Nonlinear Mappings of Monotone Type I. Classification and Degree Theory, Preprint no. 2/88, Mathematics, University of Oulu. 7. Fan, X. L. and Zhang, Q. H. Existence of Solutions for \(p(x)\)-Laplacian Dirichlet Problem, Nonlinear Analysis: Theory, Methods and Applications, 2003, vol. 52, no. 8, pp. 1843-1852. DOI: 10.1016/S0362-546X(02)00150-5. 8. Ilias, P. S. Existence and Multiplicity of Solutions of a \(p(x)\)-Laplacian Equations in a Bounded Domain, Revue Roumaine des Mathematiques Pures et Appliquees, 2007, vol. 52, no. 6, pp. 639-653. 9. Fan, X. L. and Zhao, D. On the Spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\), Journal of Mathematical Analysis and Applications, 2001, vol. 263, no. 2, pp. 424-446. DOI: 10.1006/jmaa.2000.7617. 10. Kovacik, O. and Rakosnik, J. On Spaces \(L^{p(x)}\) and \(W^{1,p(x)}\), Czechoslovak Mathematical Journal, 1991, vol. 41, pp. 592-618. 11. Zhao, D., Qiang, W. J. and Fan, X. L. On Generalizerd Orlicz Spaces \(L^{p(x)}(\Omega)\), Journal of Gansu Sciences, 1996, vol. 9, no. 2, pp. 1-7. 12. Samko, S. G. Density of \(\mathcal{C}_0^\infty (\mathbb{R}^N)\) in the Generalized Sobolev Spaces \(W^{m,p(x)}(\mathbb{R}^N)\), Doklady Akademii Nauk, 1999, vol. 369, no. 4, pp. 451-454. 13. Chang, K. C. Critical Point Theory and Applications, Shanghai, Shanghai Scientific and Technology Press, 1986. 14. Zeidler, E. Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, New York, Springer-Verlag, 1985. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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