On The Study For Singular Perturbed Elliptic Partial Differential Equations

In this paper,we mainly study several types of singular perturbed elliptic partial differential equations.The thesis consists of four chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some notations used in the whole thesis.In Chapter Two,we study the following Kirchhoff type problem:-(a+b?RNf|?u|2)?u=(1+?K(X))u2*-1,u>0 in RN,where a,b>0 are given constants,?>0 is a small parameter a-d 2*=2N/N-2,(N?3).We show that if K(x)has k critical points near which K(x)satisfies some expansion assumption,then by Lyapunov-Schmidt reduction method,we construct multi-peak solutions for ?>0 small,which concentrate at the k critical points of K(x).The result extend the work on the existence of multi-peak solutions for Schr(?)dinger equation by Cao et al.[Calc.Var.Partial Differential Equations 15,(2002)403-419].In particular,for N? 5,we can find a pair of multi-peak solutions,this is quite different from Schr(?)dinger equation.In Chapter Three,we are concerned with the following nonlinear elliptic prob-lem:-?2?u+?V(x)u=up+u2*-1,u>0 in RN,where ??R+,N?3,p?(1,2*-1)with 2*=2N/N-2,?>0 is a small parameter and V(x)is a given function.Under suitable assumptions,we prove that problem has multi-peak solutions by Lyapunov-Schmidt reduction method for sufficiently small?,which concentrate at local minimum points of potential function V(x).Moreover,we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.In Chapter Four,we are concerned with the following fractional Schr(?)dinger-Poisson system:where s ?(3/4,1),t ?(0,1),? is a positive parameter,2*s=6/3-2s is the critical Sobolev exponent.K(x)?L 6/2t+4s-3(R3),V(x)?L 3/2s(R3)and V((x)is assumed to be zero in some region of R3,which means that the problem is of the critical frequency case.In virtue of a global compactness result in fractional Sobolev space and Lusternik-Schnirelman theory of critical points,we succeed in proving the multiplicity of bound states.The results improve the results for fractional Schr(?)dinger equation by Zhang et al.[Nonlinear Anal.190(2020),111599,15 pp].