一个带约束双调和方程Navier边值问题的多解

Multiple Solutions of Navier Boundary Value Problem for a Biharmonic Equation with Constraint

讨论带约束的双调和方程{△^2U=λf(x,u)x∈Ω,1/2∫Ω|△u|^2dx=a,a>0,u|■Ω=△u|■Ω=0,其中Ω是R^N(N>4)的一个具有光滑边界的有界区域,利用变分方法证明了非线性项在某些适当假设下存在两个解,一个是正解,一个是负解。

We deal with the following problem:{△^2U=λf(x,u)x∈Ω,1/2∫Ω|△u|^2dx=a,a>0,u|■Ω=△u|■Ω=0,where Ω■R^N is a bounded domain with smooth boundary ■Ω,Δ^2 is the biharmonic operator,andλis a constant.In this paper,we prove that there exist positive and negative solutions under certain appropriate assumptions via variational method.