The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation: { ∑ | α | ⩽ m ( − 1 ) | α | D α A α ( x , ξ m ( u ) ) = G + f ( x , u ) , x ∈ Ω , D β u = 0 , x ∈ ∂ Ω , ∀ | β | ⩽ m − 1 , where Ω ⊂ R N is a bounded domain, 1 < p − ⩽ p + < N m . G is a bounded linear functional on W 0 m , p ( x ) ( Ω ) , and A α ( x , ξ m ( u ) ) , f ( x , u ) satisfy Carathéodory condition and some p ( x ) -growth conditions, respectively. We show that there exists at least one non-trivial solution for the above problem in W 0 m , p ( x ) ( Ω ) .