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

A Weighted Eigenvalue Problems Driven by both p(·)-Harmonic and p(·)-Biharmonic Operators AbstractThe existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operatorsΔp(x)2u-Δp(x)u=λw(x)|u|q(x)-2u   in  Ω,            u∈W2,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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

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

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Publisher
de Gruyter
Copyright
© 2021 Mohamed Laghzal et al., published by Sciendo
eISSN
2336-1298
DOI
10.2478/cm-2020-0011
Publisher site
See Article on Publisher Site

Abstract

AbstractThe existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operatorsΔp(x)2u-Δp(x)u=λw(x)|u|q(x)-2u   in  Ω,            u∈W2,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).

Journal

Communications in Mathematicsde Gruyter

Published: Dec 1, 2021

Keywords: Palais-Smale condition; Ljusternick-Schnirelmann; Variational methods; p (·)-biharmonic operator; p (·)-harmonic operator; Variable exponent; 58E05; 35J35; 47J10

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