315 research outputs found
Rotationally Symmetric -harmonic maps
We study a second order ordinary differential equation corresponding to
rotationally symmetric -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 and any positive number we establish the
existence of smooth functions K on with , such that the equation in 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 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 and be two different positive smooth solutions of the
equation in By a result of Gidas, Ni and Nirenberg, and are radially
symmetric above the points and , respectively. Let be a
positive -function on such that in and in , where and are disjoint non-empty open
domains in . satisfies the equation in By the same result of Gidas, Ni and Nirenberg,
in . In this paper we discuss lower bounds on
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 -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
Asymptotic properties of energy of harmonic maps on asymptotically hyperbolic manifolds
Asymptotic behavior of energy of a harmonic map defined on an asymptotically
hyperbolic manifold is considered. Using the growth of energy, we show that a
harmonic map defined on some asymptotically hyperbolic manifolds has to be
constant if the total energy is finite, or if the map approaches a point fast
enough, in terms of a defining function for the boundary.Comment: LaTeX, 17 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 in R^n and Positive Scalar Curvature
We study asymptotic behavior of positive smooth solutions of the conformal
scalar curvature equation in . 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 has a
sequence of local maxima that approach the standard spherical solution
Growth estimates on positive solutions of the equation in
We construct unbounded positive -solutions of the equation in (equipped with Euclidean metric )
such that is bounded between two positive numbers in , the
conformal metric is complete, and the volume growth of
can be arbitrarily fast or reasonably slow according to the constructions.
By imposing natural conditions on , we obtain growth estimate on the
-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
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