Multiple Positive Solutions Of Biharmonic Equations With Hardy Terms

The variational method is based on the theory of critical point,which could prove the existence,multiplicity,and approximate solution by transforming the differential equation boundary value problem into a variational problem.We study two different elliptic differential equations,namely the semilinear elliptical equation with critical exponent and the quasilinear elliptic equation with p-Laplacian operator by using the variational method in this paper.In the issue,we study the following semilinear elliptical equation with critical exponent where 2*=2N/N-4 is called critical Sobolev exponent.0 ?(?)RN(N ? 5)is a bounded smooth domain,1 ? ?<2,and0<?<?N,p=(N(N-4)/4)2.By using Ekeland's variational principle,we study the existence and multiplicity of nonzero non-negative solutions when ?take different values by establishing radial-off function.And we study the following quasilinear elliptic equation with p-Laplacian operator(?)where 0 ??(?)RN(N ? 5)is a bounded smooth domain,0<?<?N,p=((p-1)N(N-2p/p2)p and 0<q<1<?<p*-1,the p*=Np/N-2p is called critical Sobolev exponent.f(x)>0 and W(x)is a given function with the set {x??:W(x)>0} of positive measure.Using the concentration compactness principle and Nehari manifold,we obtain the existence and multiplicity of nonzero non-negative solutions and provide uniform estimates of extremal values by variational methods.