On The Existence And Multiplicity Of Solutions For Nonlinear PDE With Critical Exponent

In this paper,we mainly study the existence and multiplicity of solutions for nonlinear PDE with critical exponent.The thesis consists of five 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 Choquard equation with Kirchhoff operatorwhere a?0,b>0,??(0,N)?2?*=N+?/N-2 is the critical exponent respect to Hardy-Littlewood-Sobolev inequality and V(x)?LN/2(RN)is a given nonnegative function.By using the classical Linking theorem and global compactness theorem,we prove that equation has at least one bound state solution if ?V?LN/2 is small.More intriguingly,our result covers a novel feature of Kirchhoff problems,which is the fact that the parameter a can be zero.In Chapter Three,we study the singularly perturbed Choquard equation?2s(-?)su+V(x)u=(I?*|u|2?,s*)|u|2?,s*-2u,u?Ds,2(RN),where s?(0,1),N?3,?is a positive parameter,2?,s*=N+?/N-2s is the critical exponent respect to Hardy-Littlewood-Sobolev inequality.V(x)?LN/2s(RN)and V(x)is assumed to be zero in some region of RN,Wlwhich means 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 state solutions.In Chapter Four,we study the following fractional Schrodinger equation with critical growth(-?)su+V(x)u=|u|2s*-2u,x?RN,where s ?(0,1),N>4s,(-?)s is the fractional Laplacian operator of order s,potential function V(x):RN?R,2s*=2N/N-2s is the fractional critical Sobolev exponent.In virtue of a barycenter function,quantitative deformation lemma and Brouwer degree theory,we prove the existence and multiplicity of positive high energy solutions.Our results extend and improve the recent work on the existence of high energy solutions for fractional Schrodinger equation by Correia and Figueiredo(Calc.Var.Partial Differential Equations,58:63,(2019)).In Chapter Five,we study the singularly perturbed p-Laplacian equation-??pu+V(x)|u|p-2u=|u|p*-2u,u?D1,p(RN),where 1<p<N,p-Laplacian operator ?p:=div(|?|p-2?u),p*=Np/(N-p),? is a positive parameter,V(x)E ?LN/p(RN)and V(x)is assumed to be zero in some region of RN,which means it is of the vanishing potential case.In virtue of Lusternik-Schnirelman theory of critical points,we succeed in proving the multiplici-ty of positive solutions.This result generalizes the result for semi-linear Schrodinger equation by Chabrowski and Yang(Port.Math.57(2000),273-284).