Existence And Multiplicity Of Sign-changing Solutions For Nonlinear Elliptic Equations With Critical Index

This paper deals mainly with two classes of nonlinear elliptic equations with critical index,namely fourth-order Kirchhoff equations and Schr(?)dingerPoisson system.Variational method and critical point theory are utilized to achieve the existence and multiplicity of nodal solutions.The paper is comprised of three chapters:In Chapter One,we give an overview of the background and current research status related to the topic at home and abroad,a brief introduction to usual notations and preliminary theorems needed in other chapters,and a list of the skeleton of the paper.In Chapter Two,we address the existence of nodal solutions to the fourthorder elliptic equations of Kirchhoff type with critical growth in RN where ?2u is the biharmonic operator,2**=2N/(N-4)is the critical Sobolev exponent with 5 ?N<8,b and ? are two positive parameters.The continuous functions V(x)and f(u)satisfy certain conditions.By using a main tool of constrained minimization on Nehari manifold,we establish the existence,energy estimate and the convergence property of nodal solutions to the above equation,complementing the existing relevant results.In Chapter Three,the following Schrodinger-Poisson system with critical exponential growth are considered where ?Nu=div(|?u|N-2?u)is the N-Laplacian operator and ?,?>0.Here V(x)and f(u)are smooth functions satisfying certain hypotheses.We obtain the existence,energy estimate and the convergence property of nodal solutions to the problem in RN,with the help of constrained minimization on Nehari manifold.