Existence of solutions for a family of polyharmonic and biharmonic equations

We consider a family of polyharmonic problems of the form (−∆)mu = g(x,u) in Ω, Dαu = 0 on ∂Ω, where Ω ⊂ Rn 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 ∆2u+ (∆φ+ |∇φ|2)∆u+ 2∇φ ·∇∆u = g(x,u), where φ ∈ C2(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.