本文中我们利用移动球面法研究平面上半线性椭圆方程全局解的性质.半线性椭圆方程△u +eu|x|α = 0, (0.0.1)在Riemann 几何以及天体物理等方面有重要应用. 在高维情形, 其全局解的对称性及非存在性等问题, 已被L Nirenberg 等多位数学家广泛研究. 在低维情形,当0 < α < 4 时, x = 0 是方程(0.0.1) 的奇点, 我们证明了其满足条件lim supx→0u(x)ln |x| ≤ 0, lim infx→∞u(x)ln |x| ≥ 0 (0.0.2)的全局解u ∈ C2(R2\{0}) 关于原点对称, 从而是不存在的. 条件(0.0.2) 包含了正解的情形. 当α < 0 时, 我们证明方程(0.0.1) 满足条件(0.0.2) 第二式的全局解u ∈ C2(R2) 形如u(x) ≡ ln(ad + |x − ¯x|2 )q2 ,其中q > 0, a ≥ 0, d > 0 和¯x ∈ R2, 从而也是不存在的.我们讨论了全平面上带有奇性的半线性椭圆方程Δu +cu|x|2 + eu = 0, c > 0. (0.0.3)在高维情形, H Brezis, E Lieb 和YanYan.Li 等人研究类似的方程. 在低维情形,我们证明了方程(0.0.3) 满足条件(0.0.2) 的全局解的对称性和非存在性.最后我们利用移动球面法给出了经典的Liouville 定理的新证明.

In this thesis we study the global solutions of semilinear elliptic equationsby employing the method of moving spheres.Consider the semilinear elliptic equation△u +eu|x|α = 0, (0.0.4)which arises in many fields, such as astrophysics, chemical diffusion, Riemanniangeometry and so on.L Nirenberg well-studied the global solution of this equation, obtaining thesymmetry property and nonexistence result in high-dimensional spaces. In lowdimensionalspace, when 0 < α < 4, x = 0 is a singular point of this equation.Considering the global solution u of this equation, we establish u which satisfieslim supx→0u(x)ln |x| ≤ 0, lim infx→∞u(x)ln |x| ≥ 0 (0.0.5)is radially symmetric about the origin. Consequently, there does not exist aglobal solution satisfing the condition (0.0.5). When α < 0, we show u satisfiesthe second condition of (0.0.5) having the formu(x) ≡ ln(ad + |x − ¯x|2 )q2 ,here q > 0, a ≥ 0, d > 0 and ¯x ∈ R2. So we establish the nonexistence resultfor u.