Geodesic
A Weighted Eigenvalue Problems Driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-Biharmonic Operators
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 443-455.

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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 $W^{m,p(\cdot )}(\Omega )$.
Classification : 35J35, 47J10, 58E05
Keywords: Palais-Smale condition; Ljusternick-Schnirelmann; Variational methods; $p(\cdot )$-biharmonic operator; $p(\cdot )$-harmonic operator; Variable exponent. 
@article{COMIM_2021__29_3_a8,
     author = {Laghzal, Mohamed and Khalil, Abdelouahed El and Touzani, Abdelfattah},
     title = {A {Weighted} {Eigenvalue} {Problems} {Driven} by both $p(\cdot )${-Harmonic} and $p(\cdot )${-Biharmonic} {Operators}},
     journal = {Communications in Mathematics},
     pages = {443--455},
     publisher = {mathdoc},
     volume = {29},
     number = {3},
     year = {2021},
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     zbl = {07484379},
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     url = {http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a8/}
}
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Laghzal, Mohamed; Khalil, Abdelouahed El; Touzani, Abdelfattah. A Weighted Eigenvalue Problems Driven by both $p(\cdot )$-Harmonic and $p(\cdot )$-Biharmonic Operators. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 443-455. http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a8/