On the semilinear equation Delta(u) + k(x)u - f(x,u) = 0 on complete manifolds.

摘要

This thesis is divided into two parts. In the first part (chapter 1, 2 and 3), we consider the semilinear elliptic equation{dollar}{dollar}Delta u+k(x)u-f(x,u)=0leqno(0.1){dollar}{dollar}on a n-dimensional complete noncompact Riemannian manifold ({dollar}M,g{dollar}). In the special case that {dollar}f(x,u)=K(x)usp{lcub}p{rcub}, p={lcub}n+2over n-2{rcub},ngeq3{dollar}, this equation becomes the well known Yamabe's equation{dollar}{dollar}Delta u+k(x)u-K(x)usp{lcub}p{rcub}=0leqno(0.1)spprime{dollar}{dollar}and it is originated from the problem of prescribing scalar curvature on Riemannian manifolds. Numerous works have been done by many authors for (0.1)'.; We will study equation (0.1) with the assumption that {dollar}f(x,u) geq 0{dollar} is essentially positive and satisfies some minor growth conditions in the {dollar}u{dollar} variable. Equation (0.1) is well adapted to the super and subsolution method. In other words, the local elliptic analysis has been well understood. Our main purpose here is to provide a global analysis of the equation (0.1) and to establish a general scheme for the problem of existence and nonexistence of positive solutions of the equation (0.1).; The existence problem is essentially reduced to the existence of a positive subsolution which is easy to produce in reality if a solution ever exists. Some nonexistence results are proved by applying the maximum principle if {dollar}f(x,u){dollar} decays to zero not too fast in the {dollar}x{dollar} variable near infinity. As an example, we give sharp existence and nonexistence results of equation (0.1) on {dollar}Rsp{lcub}n{rcub},n geq 3{dollar}.; In the second part (chapter 4), we study the problem of prescribing Gaussian curvature on {dollar}Rsp2{dollar}. It is well known that a continuous function {dollar}K(x){dollar} on {dollar}Rsp2{dollar} is a conformal Gaussian curvature function if and only if there is a {dollar}Csp2{dollar} solution {dollar}u{dollar} of the nonlinear equation{dollar}{dollar}Delta u + K(x)esp{lcub}2u{rcub} = 0.leqno(0.2){dollar}{dollar}We prove that every continuous nonnegative radial function {dollar}K(x) = Ksb1(vert xvert){dollar} on {dollar}Rsp2{dollar} is a conformal Gaussian curvature function. In particular, this indicates that the 2-dimensional case (prescribing Gaussian curvature) is essentially different from the problem of prescribing scalar curvature in higher dimensions since not every positive radially symmetric function on {dollar}Rsp{lcub}n{rcub}{dollar} is a conformal scalar curvature function as is indicated by W. M. Ni in {dollar}lbrack22rbrack{dollar}.