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

Abstract

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operators\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).

Infinitely Many Weak Solutions for Problems Involving Both p(x)-Laplacian and p(x)-Biharmonic Operators

“…We have to mention that in the particular case when b 2 (x) ≡ a(x) ≡ 0, we find that our problem is identical to the problem studied in [19], this shows that our main result constitutes an important development of the main result of [19], noting that in the latter the Ljusternik-Schnireleman principle on C 1 -manifold was employed in the proof.…”