一类非线性椭圆型方程的一个Liouville定理

A Liouville Theorem for a Class of Nonlinear Elliptic Equations

设 (Mn,g)是一个 n维的完备黎曼流形 ,其 Ricci曲率满足 Ric M(x)≥ - A(1 +r2 (x) ln2 (2+r(x) ) ) ,其中 A是非负常数 ,r(x)表示点 x∈ M到某固定点 x0 ∈ M的测地距离 .则 M上方程 Δu+Su+Kuα=0在下述条件“ (i)在 M上 S≤ 0 ;(ii)在 M上 K<0且有常数 a>0使在一个紧集之外 K≤ - a2 ;(iii)常数 α>1”下的 C2 -非负解只有零解 .

Let \$(M\+n, g)\$ be a complete Riemannian manifold of dimention n with Ricci curvature \$\%Ric\%\-M(x)≥-A(1+r\+2(x) ln \+2(2+r(x))),\$ where A is a nonnegative constant and \$r(x)\$ is the geodesic distance from \$x\$ to some fixed point \$x\-0∈M.\$ In this paper, we show that there is no nonnegative \$C\+2\$ solution but zero to the equation \$Δu+Su+Ku\+α=0\$ on \$M\$ under such conditions as: \$S≤0\$ on \$M; K<0\$ on \$M\$ and \$K≤-a\+2<0\$ outside some compact set for a constant \$a>0\$ and any constant \$α>1.\$