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1.
This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n?3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n?4, we give conditions on the function h to guarantee there is only one simple blow-up point. 相似文献
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Wael Abdelhedi Hichem Chtioui Mohameden Ould Ahmedou 《Annals of Global Analysis and Geometry》2009,36(4):327-362
We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur
and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of,
on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler–Lagrange
functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological
invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds. 相似文献
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Kuo-Shung Cheng Chang-Shou Lin 《Calculus of Variations and Partial Differential Equations》2000,11(2):203-231
In this paper, we consider the equation
where is a nonpositive function in . A solution u is said to be complete if the conformal metric is complete in . Let
Assuming only that , we prove that equation (0.1) possesses infinitely many complete solutions. If in addition, K is assumed to satisfy
for some positive constant m, then is also necessary for equation (0.1) to have a complete solution with finite total curvature. We are also able to classify
the solution set of equation (0.1) for a wider class of the curvature function K than those considered in [5, 6].
Received October 1, 1997 / Revised version August 10, 1999 / Published online April 6, 2000 相似文献
8.
Shinichiroh Matsuo 《Mathematische Annalen》2014,360(3-4):675-680
We solve the modified Kazdan–Warner problem of finding metrics with prescribed scalar curvature and unit total volume. 相似文献
9.
Aymen Bensouf Hichem Chtioui 《Calculus of Variations and Partial Differential Equations》2011,41(3-4):455-481
In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n????5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler?CHopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature. 相似文献
10.
Given a smooth function
, we call H-bubble a conformally immersed surface in
3 parametrized on the sphere
2 with mean curvature H at every point. We prove that if
is a nondegenerate stationary point for H with
, then there exists a curve
of embedded H-bubbles, defined for large, which become round and concentrate at
as
. Also the case of topologically stable extremal points for H is considered.Work supported by M.U.R.S.T. progetto di ricerca Metodi variazionali ed equazioni differenziali nonlineari (cofin. 2002)
Mathematics Subject Classification (2000):53A10 (49J10) 相似文献
11.
José F. Escobar 《Journal of Functional Analysis》2003,202(2):424-442
Let (Mn,g) be a compact manifold with boundary with n?2. In this paper we discuss uniqueness and non-uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of M. 相似文献
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Marcos Dajczer Pedro A. Hinojosa Jorge Herbert de Lira 《Calculus of Variations and Partial Differential Equations》2008,33(2):231-248
It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian
manifolds.
M. Dajczer was partially supported by Procad, CNPq and Faperj. P. A. Hinojosa was partially supported by PADCT/CT-INFRA/CNPq/MCT
Grant #620120/2004-5. J. H. de Lira was partially supported by CNPq and Funcap. 相似文献
13.
Hongjing Pan Ruixiang Xing 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7437-7445
We prove the nonexistence of solutions for a prescribed mean curvature equation when p?1 and the positive parameter λ is small. The result extends theorems of Narukawa and Suzuki, and Finn, from the case of n=2,p=1 to all n?2,p?1. Moreover, our proof is very simple and the result is not limited to positive (and negative) solutions. We also show that a similar result for positive solutions is still true if |u|p−1u is replaced by the exponential nonlinearity eu−1. 相似文献
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We study a quasilinear elliptic equation in the unit ball ofℝ
m
. Using this result we get the existence of graphs with prescribed curvature on hyperbolic spacesℍ
m
inℍ
m
×ℝ. 相似文献
16.
Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies and where \(\delta _{p}\) designates Dirac mass at the point \(p\).
相似文献
$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$
$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$
$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$
17.
Dr. Helmut Reckziegel 《manuscripta mathematica》1974,13(1):69-71
In a recent paper [2] K. Nomizu has shown that a natural analogue of an n-sphere in an arbitrary Riemannian manifold is an n-dimensional umbilical submanifold with non-zero parallel mean curvature vector, which he calls extrinsic sphere sometimes. This note is concerned with the question whether extrinsic spheres have a special topological or differentiable feature. 相似文献
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In this paper, we construct a family of three-dimensional asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to −6. The manifolds we construct can be arbitrarily close to anti-de Sitter-Schwarzschild manifolds at infinity. Hence, the mass of our manifolds can be very large or very small. The main arguments we use in this paper are gluing methods which are used by Miao in (Proc Am Math Soc 132(1):217–222, 2004). 相似文献