The Existence Of Multi-bump Solutions And Positive Solutions For Fractional Kirchhoff Type Problem

in this dissertation,we study the singular perturbation problem of the following fractional Kirchhoff equation where(-Δ)s is the fractional Laplace operator,a,b>0,s∈(0,1)and 2<p ≤2s*:=2N/N-2s,ε>0 is sufficiently small.On the one hand,fractional Kirchhoff equations are a class of important nonlocal elliptic equations,whose nonlocality comes from fractional Laplace operators.On the other hand,it comes from the Kirchhoff nonlocal term b ∫RN|(-Δ)s/2u|2dx(-Δ)s u.The interaction of these two nonlocalities makes it difficult to study such elliptic equations using classical variational methods.In recent years,the application of finite dimensional reduction and perturbation methods to study the existence of solutions for elliptic equations has received widespread attention.In[37,43,45],the uniqueness and non degeneracy of positive solutions for fractional Kirchhoff equations are established.On this basis,the existence of multiple envelope solutions for fractional singularly perturbed Kirchhoff problems is proved by using Lyapunov-Schmidt reduction,and the existence of positive solutions for a class of fractional Kirchhoff equations with critical Sobolev exponents is established by using perturbation methods.