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37
On the CaffarelliKohnNirenberg Inequalities: Sharp Constants, Existence (and Nonexistence), and Symmetry of Extremal Functions
"... Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]: # # # < a <(N 2)/2, a 2N/(N a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of ..."
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Cited by 100 (5 self)
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Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]: # # # < a <(N 2)/2, a 2N/(N a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case 2 are also given. c 2001 John Wiley & Sons, Inc. 1
Blowing up and desingularizing constant scalar curvature Kähler manifolds
"... Abstract. This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact ..."
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Cited by 33 (1 self)
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Abstract. This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kähler metrics.
Some new entire solutions of semilinear elliptic . . .
"... We prove existence of a new type of positive solutions of the semilinear equation −∆u+u = up on Rn, where 1 < p < n+2n−2. These solutions are bounded, but do not tend to zero at infinity. Indeed, they decay to zero away from three halflines with a common origin, and their asymptotic profile ..."
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Cited by 17 (1 self)
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We prove existence of a new type of positive solutions of the semilinear equation −∆u+u = up on Rn, where 1 < p < n+2n−2. These solutions are bounded, but do not tend to zero at infinity. Indeed, they decay to zero away from three halflines with a common origin, and their asymptotic profile is periodic along these halflines.
SINGULAR YAMABE METRICS AND INITIAL DATA WITH EXACTLY KOTTLER–SCHWARZSCHILD–DE SITTER ENDS II. GENERIC METRICS
"... Abstract. We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides timesymmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with exactly Kott ..."
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Cited by 14 (3 self)
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Abstract. We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides timesymmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with exactly Kottler–Schwarzschild–de Sitter ends, extending the results in [5]. 1.
Complex antiselfdual instantons and Cayley submanifolds
"... Let M be a CalabiYau manifold of complex dimension 4 with symplectic form ω and complex volume form θ. A connection A on a vector bundle ..."
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Cited by 10 (0 self)
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Let M be a CalabiYau manifold of complex dimension 4 with symplectic form ω and complex volume form θ. A connection A on a vector bundle
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE σkYAMABE EQUATION NEAR ISOLATED SINGULARITIES
, 2009
"... σkYamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In [38] YanYan Li proved that an admissible solution with an isolated singularity at 0 ∈ R n to the σkYamabe equation is asymptotically radially symmetric. In this work we prove that an admissible ..."
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Cited by 9 (4 self)
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σkYamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In [38] YanYan Li proved that an admissible solution with an isolated singularity at 0 ∈ R n to the σkYamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at 0 ∈ R n to the σkYamabe equation is asymptotic to a radial solution to the same equation on R n \ {0}. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σk curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.
Asymptotic behavior of positive solutions of the equation ∆g u+ Ku p = 0 in a complete Riemannian manifold and positive scalar curvature
 Comm. Partial Differential Equations
, 1999
"... We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in IR 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 radi ..."
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Cited by 7 (6 self)
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We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in IR 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 IR × S n−1 has a sequence of local maxima that approach the standard spherical solution.
Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity
 Proc. Amer. Math. Soc
"... Abstract. We study the conformal scalar curvature problem k(x)u n+2 n−2 ≤−∆u≤u n+2 n−2 in R n,n≥3 where k: R n → (0, 1] is a continuous function. We show that a necessary and sufficient condition on k for this problem to have C 2 positive solutions which are arbitrarily large at ∞ is that k be less ..."
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Cited by 5 (3 self)
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Abstract. We study the conformal scalar curvature problem k(x)u n+2 n−2 ≤−∆u≤u n+2 n−2 in R n,n≥3 where k: R n → (0, 1] is a continuous function. We show that a necessary and sufficient condition on k for this problem to have C 2 positive solutions which are arbitrarily large at ∞ is that k be less than 1 on a sequence of points in R n which tends to ∞. 2000 Mathematics Subject Classification. Primary 35J60, 53C21 1