1. bookAHEAD OF PRINT
Journal Details
License
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English
access type Open Access

A Weighted Eigenvalue Problems Driven by both p(·)-Harmonic and p(·)-Biharmonic Operators

Published Online: 10 Aug 2020
Page range: -
Received: 11 Aug 2019
Accepted: 23 Sep 2019
Journal Details
License
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English
Abstract

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operators

Δp(x)2u-Δp(x)u=λw(x)|u|q(x)-2uinΩ,uW2,p()(Ω)W0-1,p()(Ω),\eqalign{& \Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u = \lambda w\left( x \right){\left| u \right|^{q\left( x \right) - 2}}u\,\,\,{\rm{in}}\,\,\Omega {\rm{,}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,u \in {W^{2,p\left( \cdot \right)}}\left( \Omega \right) \cap W_0^{ - 1,p\left( \cdot \right)}\left( \Omega \right), \cr}

is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp(·)(Ω) and Wm,p(·)(Ω).

Keywords

MSC 2010

[1] T.A. Bui: W1,p(·) estimate for renormalized solutions of quasilinear equations with measure data and Reifenberg domains. Advances in Nonlinear Analysis 7 (4) (2018) 517–533.Search in Google Scholar

[2] M. Cencelj, V.D. Răadulescu, D.D. Repovš: Double phase problems with variable growth. Nonlinear Analysis 177 (2018) 270–287.Search in Google Scholar

[3] L. Diening, P. Harjulehto, P. Hästö, M. Ruzicka: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, Springer (2011).Search in Google Scholar

[4] D. Edmunds, J. Rákosník: Sobolev embeddings with variable exponent. Studia Mathematica 143 (3) (2000) 267–293.Search in Google Scholar

[5] A. El Khalil, M.D. Morchid Alaoui, A. Touzani: On the Spectrum of problems involving both p(x)-Laplacian and P (x)-Biharmonic. Advances in Science, Technology and Engineering Systems Journal 2 (5) (2017) 134–140.Search in Google Scholar

[6] X. Fan, X. Han: Existence and multiplicity of solutions for p(x)-Laplacian equations in Dirichlet problem in 𝕉N. Nonlinear Analysis: Theory, Methods & Applications 59 (1–2) (2004) 173–188.Search in Google Scholar

[7] X.L. Fan, X. Fan: A Knobloch-type result for p(t)-Laplacian systems. Journal of mathematical analysis and applications 282 (2) (2003) 453–464.Search in Google Scholar

[8] X.L. Fan, X. Fan: A Knobloch-type result for p(t)-Laplacian systems. Journal of mathematical analysis and applications 282 (2) (2003) 453–464.Search in Google Scholar

[9] X.L. Fan, Q.H. Zhang: Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Analysis: Theory, Methods & Applications 52 (8) (2003) 1843–1852.Search in Google Scholar

[10] K. Kefi, V.D. Răadulescu: On a p(x)-biharmonic problem with singular weights. Zeitschrift für angewandte Mathematik und Physik 68 (80) (2017) 1–13.Search in Google Scholar

[11] A. Scapellato: Regularity of solutions to elliptic equations on Herz spaces with variable exponents. Boundary Value Problems 2019 (1) (2019) 1–9.Search in Google Scholar

[12] M. Mihăailescu, V. Răadulescu: A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462 (2073) (2006) 2625–2641.Search in Google Scholar

[13] V.D. Răadulescu: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Analysis: Theory, Methods & Applications 121 (2015) 336–369.Search in Google Scholar

[14] V.D. Răadulescu: Isotropic and anisotropic double-phase problems: old and new. Opuscula Mathematica 39 (2) (2019) 259–279.Search in Google Scholar

[15] V.D. Răadulescu, D.D. Repovš: Partial differential equations with variable exponents: variational methods and qualitative analysis. Monographs and Research Notes in Mathematics, CRC press (2015).Search in Google Scholar

[16] M. Růžička: Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, 1748, Springer Science & Business Media (2000).Search in Google Scholar

[17] A. Szulkin: Ljusternik-Schnirelmann theory on C1-manifolds. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 5 (2) (1988) 119–139.Search in Google Scholar

[18] A. Zang, Y. Fu: Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces. Nonlinear Analysis: Theory, Methods & Applications 69 (10) (2008) 3629–3636.Search in Google Scholar

[19] E. Zeidler: Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators. Springer (1990). Translated from the German by the author and Leo F. BoronSearch in Google Scholar

[20] Q. Zhang, V.D. Răadulescu: Double phase anisotropic variational problems and combined e ects of reaction and absorption terms. Journal de Mathématiques Pures et Appliquées 118 (2018) 159–203.Search in Google Scholar

[21] V.V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 50 (4) (1986) 675–710.Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo