Rotationally Symmetric pp-harmonic maps

We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investigated.Comment: LaTex, 41 page

Blow-up solutions of nonlinear elliptic equations in R^n with critical exponent

For an integer $n \ge 3$ and any positive number $\epsilon$ we establish the existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le \epsilon$, such that the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n \setminus \{0 \}$ has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some cases K can be extended as a Lipschitz function on ${\R}^n.$ These provide counter-examples to a conjecture of C.-S. Lin when n > 4, and Taliaferro's conjecture.Comment: 27 page

Combining solutions of semilinear partial differential equations in R^n with critical exponent

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\Delta u + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the points $\xi_1$ and $\xi_2$, respectively. Let $u$ be a positive $C^2$-function on $R^n$ such that $u = u_1$ in $\Omega_1$ and $u = u_2$ in $\Omega_2$, where $\Omega_1$ and $\Omega_2$ are disjoint non-empty open domains in ${\R}^n$. $u$ satisfies the equation $\Delta u + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n.$ By the same result of Gidas, Ni and Nirenberg, $K \not\equiv 1$ in $R^n$. In this paper we discuss lower bounds on $\displaystyle{\sup_{\R^n} |K - 1|} .$ Relation with decay estimates at the isolated singularity via the Kelvin transform is also considered.Comment: 35 page

Rotationally Symmetric F-harmonic Maps Equations

We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.Comment: LaTex, 21 page

Conformal Deformation of Warped Products and Scalar Curvature Functions on Open Manifolds

We discuss conformal deformation and warped products on some open manifolds. We discuss how these can be applied to construct Riemannian metrics with specific scalar curvature functions.Comment: 25 pages, LaTex forma

Conformal Scalar Curvature Equations in Open Spaces

The article contains a brief description on the study of conformal scalar curvature equations, and discusses selected topics and questions concerning the equations in open spaces.Comment: 28 page

Asymptotic Behavior of Positive Solutions of the Equation Δu+Ku(n+2)/(n−2)=0\Delta u + K u^{(n + 2)/(n - 2)} = 0 in R^n and Positive Scalar Curvature

We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in ${\bf R}^n$. We consider the case when the scalar curvature of the conformal metric is bounded between two positive numbers outside a compact set. It is shown that the solution has slow decay if the radial change is controlled. For a positive solution with slow decay, the corresponding conformal metric is found to be complete if and only if the total volume is infinite. We also determine the sign of the Pohozaev number in some situations and show that if the Pohozaev is equal to zero, then either the solution has fast decay, or the conformal metric corresponding to the solution is complete and the corresponding solution in ${\bf R} \times S^{n - 1}$ has a sequence of local maxima that approach the standard spherical solution

Growth estimates on positive solutions of the equation Δu+Kun+2n−2=0\Delta u + K u^{{n + 2}\over {n - 2}} = 0 in Rn{\R}^n

We construct unbounded positive $C^2$-solutions of the equation $\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in ${\R}^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in ${\R}^n$, the conformal metric $g = u^{4/(n - 2)} g_o$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the $L^{2n/(n - 2)}$-norm of the solution and show that it has slow decay.Comment: 15 page

Construction of Blow-up Sequence for the Conformal Scalar Curvature Equation on S^n. I, II, and Appendix

Using the Lyapunov-Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on S^n, (n > 5), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have C^{n - 1, \beta} smoothness.Comment: Part I, 32 pages; Part II, 31 pages; Appendix, 32 page

Uniqueness of Positive Solutions of the Conformal Scalar Curvature Equation and Applications to Conformal Transformations

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We also study uniqueness of complete positive solutions and radial solutions.Comment: LaTeX, 16 page