Multiplicity results for p(x)-biharmonic equations with nonlinear boundary conditions.

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a p (x) -biharmonic operator { Δ p (x) 2 u = λ f (x) | u | q (x) − 2 u , x ∈ Ω , ∂ (| Δ u | p (x) − 2 Δ u) ∂ n = g (x) | u | r (x) − 2 u , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded domain, Δ p (x) 2 u = Δ (| Δ u | p (x) − 2 Δ u) is the operator of fourth order called the p (x) -biharmonic operator, p (x) , q (x) , r (x) ∈ C (Ω ¯) , and f ∈ C (Ω ¯) , g ∈ C (∂ Ω) are non-negative weight functions with compact support in Ω. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue–Sobolev spaces L p (x) (Ω) and W m , p (x) (Ω). [ABSTRACT FROM AUTHOR]