Normalized Solutions For Some Fractional Kirchhoff Equations

Kirchhoff equations have been widely studied by many scholars in recent decades.Kirchhoff equations with fractional Laplacian operator frequently appear in many different research fields,and have many applications in fractional quantum mechanics,physics and chemistry,obstacle problems,optimization and finance,conformal geometry and minimal surfaces and so on.This thesis mainly uses variational methods and mountain pass theorem to study the existence of normalized solutions for a class of fractional Kirchhoff type equations.The thesis consists of the following four chapters:In chapter 1,we introduce the research background and research status of normalized solutions for fractional Kirchhoff type equations,and give the preliminary knowledge required for this thesis.In chapter 2,we study the sharp existence of constrained minimizers for the fractional Kirchhoff equation with mass subcritical and critical nonlinearity:(a+b∫R3|(-Δ)s/2u|2dx)(-Δ)su=λu+|u|p-2u,x∈R3,satisfying the normalization constraint∫R3 u2=c2 where a,b,c>0,s∈(3/4,1),p ∈(2,6+8s/3],and λ∈R.The fractional Laplacian operator(-Δ)s can be defined by(-Δ)sv(x)=CsP.V.∫R3 v(x)-v(y)/|x-y|3+2s dy=Cs(?)∫R3\Bε(x)v(x)-v(y)/|x-y|3+2s dy for v ∈ S(R3),where S(R3)is the Schwartz space of rapidly decaying C∞ function,Bε(x)denote an open ball of radius ε centered at x and the normalization constant Cs=(∫R3 1-cos(ζ1)/|ζ|3+2s)-1.For 2<p≤ 6+8s/3,we obtain the sharp existence and nonexistence of global constraint minimizers,and thus the existence and nonexistence of a normalized solution.In Chapter 3,we give the existence of normalized ground state solution for the fractional Kirchhoff equation with mass supercritical nonlinearity:(a+b∫R3|(-Δ)s/2u|2dx)(-Δ)su=λu+g(u),x∈R3,satisfying the normalization constraint∫R3 u2=c2 where a,b,c>0,s∈(3/4,1)and λ∈R,(-Δ)s denotes the fractional Laplacian operator.Under fairly general assumptions on the nonlinearity g,we can prove the existence of a ground state normalized solution for any given c>0.In Chapter 4,we consider mainly the existence of normalized ground states to the Kirchhoff type problem with critical or supercritical nonlinearity where s ∈(3/4,1),a,b,c>0,6+8s/3<q<2s*,p≥ 2s*(2s*=6/3-2s),μ>0 and λ∈R as a Langrange multiplier.By combining an appropriate truncation argument with Moser iteration method,we prove that the existence of normalized solutions for the above equation when the parameter μ is sufficiently small.