Existence of solutions for a family of polyharmonic and biharmonic equations

We consider a family of polyharmonic problems of the form ( − Δ ) m u = g ( x , u ) in Ω , D α u = 0 on ∂ Ω , where Ω ⊂ ℠n is a bounded domain, g ( x , ⋅ ) ∈ L ∞ ( Ω ) , and | α | < m . By using the fibering method, we obtain some results about the existence of solution and its multiplicity under certain assumptions on g . We also consider a family of biharmonic problems of the form Δ 2 u + ( Δ ϕ + | ∇ ϕ | 2 ) Δ u + 2 ∇ ϕ ⋅ ∇ Δ u = g ( x , u ) , where ϕ ∈ C 2 ( Ω ¯ ) , and Ω , g , and the boundary condition are the same as above. For this problem, we prove the existence and multiplicity of solutions too.