International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3405-3417
doi:10.1155/IJMMS.2005.3405

Existence of solutions for a family of polyharmonic and biharmonic equations

Abstract

We consider a family of polyharmonic problems of the form (−Δ)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 Δ2u+(Δϕ+|∇ϕ|2)Δu+2∇ϕ⋅∇Δu=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.