Existence Results About Gaussian Curvature Equations And Systems

This thesis is devoted to the existence results for Liouville equations and Liouville systems.We will focus on the following parts:(1)the existence of nontrivial solutions to the mean field equations;(2)the existence results to Liouville equations with some singular soruces;(3)the stable solutions and the existence results of finite Morse index solutions to singular Liouville equations and Toda systems;(4)the existence of normal solutions for sign-changing Q-curvature equations on R4.This paper is organized as five chapters.In the first chapter,we recall the geometric background and the present progress of the existence results for Liouville equations and systems with exponential nonlinearity.The main work of this paper are concluded as well.In the second chapter.we consider the mean field equation α/2Δu+eu-1=0 on S2.We show that under some technical conditions,u has to be constantly zero for 1/3≤α<1.In particular,this is the case if u(x)=-u(-χ)and u is odd symmetric about a plane.In the cases u(x)=-u(-χ)with 1/3≤α<1 and u(x)=u(-χ)with 1/4≤α<1,we analyze the additional symmetries of the nontrivial solution in detail.In the third chapter,we consider the following Liouville system:where(?)β1,l=(?)β2,l.We show the existence of global branches of non-radial solutions with m=1,bifurcating from v1(x)=v2(x)=-2αlog|χ-P1|+log(64(1-α)2)/((2+μ)(8+|χ-P1|2(1-α))2)at the valuesμ=-2(n2+n-2)/(n2+n+2)for some special n∈N andβ1,1=β2,1＝α.We also show the existence and non-existence results with m≥3,under some suitable assumptions on β1,l,β2,l forι=1,...,m.In the forth chapter,We consider finite Morse index solutions to the non-local Glandfand Liouville equation(-Δ)su=|x|-αeu in Rn,for every s∈(0,1)and n>2s with 2s>α.We prove that there is no stable solution of the form τ(θ)-(2s-α)logr if(2s-α)/2sΓ(n/2)Γ(1+s)/(Γ((n-2s)/2))≥Γ2((n+2s)/4)/Γ2((n-2s)/4).There must be no finite Morse index solution,if the singular solution is unstable with u=(2s-α)log|χ|+log(2s-α)/2sAn,s,An,s=22sΓ(n/2)Γ(1+s)/Γ((n-2s)/2).We also consider the fractional Toda system with a singularity,as following(-Δ)sfi=|χ|-αie-(fi+1-fi)-|x|(-αi-1)e-(fi-fi-1)in Rn,withf0=-∞ and fQ+1=+∞ and i=1,…,Q.We prove the same results,i.e.,there is no stable solution of the form fi(r,θ)=ψi(θ)+(2i-Q-1)slogr-(?)αj-1logr if Moreover,there must be no finite Morse index solution,if the singular solution is unstable.In the fifth chapter,we consider the prescribed Q-curvature equation with a singularityΔ2u=|χ|-αK(x)e4u in R4,Λ=∫R4|χ|-αK(x)e4udχ<∞,for0<α<4.First,we prove that all solutions are radially symmetric for 0<α<4,K>0 constant and Λ=16π2(1-α/4).Next,some blow-up results have been studied by using a classification result for K=const>0and0<α<4.Finally,we show that equation with K(x)＝(1-|x|p)and 0<a<4 has normal solutions(namely solutions which can be written in integral form)if and only if p∈(0,4-α)and 8π2(1+(p-α)/4)≤Λ<16π2(1-α/4).