Existence of solutions for a family of polyharmonic and biharmonic equations

M. Hesaaraki, B. Raessi
2005 International Journal of Mathematics and Mathematical Sciences  
We consider a family of polyharmonic problems of the form(−Δ)mu=g(x,u)inΩ,Dαu=0on∂Ω, whereΩ⊂ℝnis 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 ong. We also consider a family of biharmonic problems of the formΔ2u+(Δϕ+|∇ϕ|2)Δu+2∇ϕ⋅∇Δu=g(x,u), whereϕ∈C2(Ω¯), andΩ,g, and the boundary condition are the same as above. For this problem, we prove the existence and multiplicity of solutions too.
doi:10.1155/ijmms.2005.3405 fatcat:glm7pxpajzavrj2smuel65bnuy