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
Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran
Received 17 April 2005; Revised 28 September 2005
Copyright © 2005 M. Hesaaraki and B. Raessi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.