Applications Of The Finite Dimensional Reduction Method In Elliptic Partial Differential Equations

In this paper,we mainly study applications of the finite dimensional reduction method in nonlinear elliptic partial differential equations.The thesis consists of five chapters:In Chapter One,we outline the background of the issues studied in this paper and the current status of research at home and abroad,and briefly introduce the main work of this paper,the relevant preparatory knowledge and some commonly used marks.In Chapter Two,we introduce the basic principle of the finite dimensional reduction method and some applications in elliptic problems.In Chapter Three,we devote to studying the model of Bose-Einstein condensates with attractive interaction,and it can be described by Gross-Pitaevskii energy functional with L2-constraint for the mass.Recently,the solutions of this model concentrated at several points have been widely considered.However,it does not seem to have the result on solutions concentrating at a high dimensional subset.We show that the existence of radial solutions of the model concentrating on spheres under suitable conditions by using modified finite dimensional reduction and blow-up analysis based on Pohozaev identity.Also we would like to point out that this concentration phenomenon is quite different from those of classical nonlinear Schrodinger equations.In Chapter Four,we study the following prescribed scalar curvature problem-△u=V(|y’|,y")u(N+2)/(N-2),u>0,u∈D1,2(RN),where(y’,y")∈R3×RN-3,V(|y’|,y")is a non-negative bounded function in R+×RN-3.We can construct new types of infinitely many solutions by using modified finite dimensional reduction method and local Pohozaev identities under the assumptions that N≥5,V(r,y")has a stable critical point(r0,y"0)with r0>0 and V(r0,y"0)>0.Particularly,in one of these cases,the new solutions can be concentrated at a pair of points with symmetry about the origin.It is noticeable to mention that Li([96])proved that this clustering kind of blow-up phenomenon cannot occur when the potential function V(y)just has isolated critical points.In Chapter Five,we revisit the well known prescribed scalar curvature problem-△u=(1+εK(x)u2*-1,u>0,u∈D1,2(RN),where N≥5,2*=2N/(N-2),ε>0 and K(x)∈C1(RN)∩L∞(RN).It is known that there are a number of results related to the existence of solutions concentrating at the isolated critical points of K(x).However,if K(x)has non-isolated critical points with different degenerate rate along different directions,whether there exist solutions concentrating at these points is still an open problem.We give a certain positive answer to this problem via applying a blow-up argument based on local Pohozaev identities and the modified finite dimensional reduction method when the dimension of critical point set of K(x)ranges from 1 to N-1 under some suitable conditions,which generalizes some results in[27,96,97].