Existence And Multiple Solutions Of Normalized Solutions Of Several Classes Of Nonlinear Elliptic Equations

In this master dissertation,we mainly consider three types of nonlinear elliptic equations(systems),including Schrodinger-Poisson equations,p-Laplacian equations and Schrodinger systems.Using variational methods,we study the existence and multiplicity of normalized solutions for the above three equations.This paper is divided into the following four chapters:In Chapter 1,we firstly introduce the physical background and research significance of these three types of equations.Then we summarize the research status and progress at home and abroad and list some basic definitions and theorems we need in this paper.Finally,we give the main results and general idea of this paper.In Chapter 2,we mainly study the following Schrodinger-Poisson equation with square root terms:where the mass c>0,the parameter λ∈R is a Lagrange multiplier.By showing the corresponding energy functional is bounded below,we obtain a global minimizer which is a normalized ground state solution of the equation.In Chapter 3,we investigate the p-Laplacian equation under supercritical conditions:where 1<p<N,λ∈ R,p<q<p*:=Np/(N-p),r=2 or r=p and V(x)is a potential satisfying positive infinity at infinity.Since the energy functional is no longer bounded below in the supercritical case,it is not possible to find a global minimizer.By constructing a new constraint space,we prove the existence of a local minimizer,which is then shown to be a ground state solution of the equation.In addition,by constructing a mountain pass geometry,we prove that there is a mountain pass solution to this equation,and prove whether the energy of the solution is positive or negative.In the last chapter,we consider the following Schrodinger system:where s>2,r1,r2>1,μ1>0,μ2>0 and β<0.V1(x)and V2(x)are potential functions and the unknown parameters λ1,λ2 are Lagrange multipliers.We study the existence and multiplicity of normal solutions of systems depending on s and r1,r2.