This thesis is divided into two parts. In the first part (chapter 1, 2 and 3), we consider the semilinear elliptic equation$$\Delta u+k(x)u-f(x,u)=0\leqno(0.1)$$on a n-dimensional complete noncompact Riemannian manifold ($M,g$). In the special case that $f(x,u)=K(x)u\sp{p},\ p={n+2\over n-2},n\geq3$, this equation becomes the well known Yamabe's equation$$\Delta u+k(x)u-K(x)u\sp{p}=0\leqno(0.1)\sp\prime$$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 $f(x,u)\ \geq\ 0$ is essentially positive and satisfies some minor growth conditions in the $u$ 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 $f(x,u)$ decays to zero not too fast in the $x$ variable near infinity. As an example, we give sharp existence and nonexistence results of equation (0.1) on $R\sp{n},n\ \geq\ 3$ In the second part (chapter 4), we study the problem of prescribing Gaussian curvature on $R\sp2$. It is well known that a continuous function $K(x)$ on $R\sp2$ is a conformal Gaussian curvature function if and only if there is a $C\sp2$ solution $u$ of the nonlinear equation$$\Delta u\ +\ K(x)e\sp{2u} = 0.\leqno(0.2)$$We prove that every continuous nonnegative radial function $K(x)\ =\ K\sb1(\vert x\vert)$ on $R\sp2$ 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 $R\sp{n}$ is a conformal scalar curvature function as is indicated by W. M. Ni in $\lbrack22\rbrack$ / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_23581 |
| Date | January 1995 |
| Contributors | Wu, Sanxing (Author), Yang, Dagang (Thesis advisor) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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