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Keywords:
Palais-Smale condition; Ljusternick-Schnirelmann; Variational methods; $p(\cdot )$-biharmonic operator; $p(\cdot )$-harmonic operator; Variable exponent.
Palais-Smale condition; Ljusternick-Schnirelmann; Variational methods; $p(\cdot )$-biharmonic operator; $p(\cdot )$-harmonic operator; Variable exponent.
Summary:
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 $W^{m,p(\cdot )}(\Omega )$.
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