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

Mohamed Laghzal; Abdelouahed El Khalil; Abdelfattah Touzani

Communications in Mathematics (2021)

  • Volume: 29, Issue: 3, page 443-455
  • ISSN: 1804-1388

Abstract

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The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p ( · ) -Harmonic and p ( · ) -biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u ∈ W 2 , p ( · ) ( Ω ) ∩ W 0 1 , p ( · ) ( Ω ) , is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces L p ( · ) ( Ω ) and W m , p ( · ) ( Ω ) .

How to cite

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Laghzal, Mohamed, Khalil, Abdelouahed El, and Touzani, Abdelfattah. "A Weighted Eigenvalue Problems Driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-Biharmonic Operators." Communications in Mathematics 29.3 (2021): 443-455. <http://eudml.org/doc/298172>.

@article{Laghzal2021,
abstract = {The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-biharmonic operators \begin\{gather*\} \Delta \_\{p(x)\}^2 u-\Delta \_\{p(x)\}u=\lambda w(x)|u|^\{q(x)-2\}u \quad \text\{in \} \Omega ,\\ u\in W^\{2,p(\cdot )\}(\Omega )\cap W\_0^\{1,p(\cdot )\}(\Omega )\,, \end\{gather*\} is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^\{p(\cdot )\}(\Omega )$ and $W^\{m,p(\cdot )\}(\Omega )$.},
author = {Laghzal, Mohamed, Khalil, Abdelouahed El, Touzani, Abdelfattah},
journal = {Communications in Mathematics},
keywords = {Palais-Smale condition; Ljusternick-Schnirelmann; Variational methods; $p(\cdot )$-biharmonic operator; $p(\cdot )$-harmonic operator; Variable exponent},
language = {eng},
number = {3},
pages = {443-455},
publisher = {University of Ostrava},
title = {A Weighted Eigenvalue Problems Driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-Biharmonic Operators},
url = {http://eudml.org/doc/298172},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Laghzal, Mohamed
AU - Khalil, Abdelouahed El
AU - Touzani, Abdelfattah
TI - A Weighted Eigenvalue Problems Driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-Biharmonic Operators
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 443
EP - 455
AB - The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-biharmonic operators \begin{gather*} \Delta _{p(x)}^2 u-\Delta _{p(x)}u=\lambda w(x)|u|^{q(x)-2}u \quad \text{in } \Omega ,\\ u\in W^{2,p(\cdot )}(\Omega )\cap W_0^{1,p(\cdot )}(\Omega )\,, \end{gather*} is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(\cdot )}(\Omega )$ and $W^{m,p(\cdot )}(\Omega )$.
LA - eng
KW - Palais-Smale condition; Ljusternick-Schnirelmann; Variational methods; $p(\cdot )$-biharmonic operator; $p(\cdot )$-harmonic operator; Variable exponent
UR - http://eudml.org/doc/298172
ER -

References

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