International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3405-3417

Existence of solutions for a family of polyharmonic and biharmonic equations

M. Hesaaraki and B. Raessi

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.


We consider a family of polyharmonic problems of the form (Δ)mu=g(x,u) in Ω, Dαu=0 on Ω, where Ωn 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+2ϕΔu=g(x,u), where ϕC2(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.