Abstract

This work deals with the study of a class of nonlocal Navier boundary value problems involving the degenerate $p(x)$-biharmonic operator with a potential term \mbox{$q(x)$-Hardy} \begin{align*} \begin{cases} \Delta( \omega (|\Delta u |^{p(x)}) |\Delta u |^{p(x)- 2}\Delta u ) - \lambda \frac{ |u|^{q(x)- 2}u}{{\delta(x)}^{2q(x)}}= \mu \vartheta(x)|u|^{q(x)- 2}u\bigg(\int_{\Omega}\frac{\vartheta(x)}{q(x)}| u|^{q(x)}dx\bigg)^{r} & \mbox{in}\ \Omega, \\ u = \Delta u =0, & \mbox{on}\ \partial \Omega. \end{cases} \end{align*} In this new setting, our objectif is to extend the results obtained in the paper [M. Laghzal, A. El Khalil, M. D. Morchid Alaoui, A. Touzani, Eigencurves of the $p(\cdot)$-biharmonic operator with a Hardy-type term potential, Moroccan J. Pure Appl. Anal., 6(2) (2020), 198--209] for the nonhomogeneous case $p(x) \neq q(x),$ where $ \vartheta$ is a weight function. The main results are established by using the variational method and min-max arguments based on Ljusternik-Schnirelmann theory on $C^1$ manifoleds [A. Szulkin, Schnirelmann theory on $C^1$-manifolds, Ann. Inst. Henri Poincaré C, Anal. Non Linéaire, 5(2) (1988), 119--139]. A direct characterization of the principal curve (first one) is provided.

Keywords: Degenerate $p(x)$-biharmonic operator; variational methods; Ljusternik-Schnirelman; nonlinear eigenvalue problems; $q(x)$-Hardy's inequality.

MSC: 35J70, 35J35, 35J75, 58J05

DOI: 10.57016/MV-LqLI2036

Pages:  1--18