Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent

Abstract

In this article, we consider the nonlinear eigenvalue problem $$displaylines{ Delta(|Delta u|^{p(x)-2}Delta u )=lambda |u|^{q(x)-2}uquad hbox{in }Omega, cr u=Delta u = 0quad hbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^{N}$ with smooth boundary and $ p, q: overline{Omega} o (1,+infty)$ are continuous functions. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekeland's variational principle.

Keywords

• Fourth order elliptic equation
• eigenvalue
• Navier condition
• variable exponent
• Sobolev space
• mountain pass theorem
• Ekeland's variational principle