On the solvability in H²(Ω) for a class of semilinear elliptic equations with dependence on the gradient

In this paper we study the existence of solutions in H²(Ω) for a class of semilinear elliptic equations with dependence on the gradient of the form - Δ u = f ( x , u , ▽ u ) + g ( x ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a bounded domain in ℝ N , N ≥ 3 , g ( x ) ∈ H l o c s ( Ω ) , f ( x , s , t ) satisfies the Hölder condition on s and t. The technique is based on the theory on a family of domains smoothly depending on a parameter in S.G. Krein‘s sense and an interative method.