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ON THE YAMABE EQUATION WITH ROUGH POTENTIALS
, 2006
"... Abstract. We study the existence of non–trivial solutions to the Yamabe equation: −∆u + a(x) = µuu  4 n−2 µ> 0, x ∈ Ω ⊂ R n with n ≥ 4, u(x) = 0 on ∂Ω under weak regularity assumptions on the potential a(x). More precisely in dimension n ≥ 5 we assume that: (1) a(x) belongs to the Lorentz spa ..."
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Abstract. We study the existence of non–trivial solutions to the Yamabe equation: −∆u + a(x) = µuu  4 n−2 µ> 0, x ∈ Ω ⊂ R n with n ≥ 4, u(x) = 0 on ∂Ω under weak regularity assumptions on the potential a(x). More precisely in dimension n ≥ 5 we assume that: (1) a(x) belongs to the Lorentz
Blowup phenomena for the Yamabe equation
 J. Amer. Math. Soc
, 2008
"... Abstract. Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A wellknown conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to ..."
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Cited by 43 (8 self)
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Abstract. Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A wellknown conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n ≥ 52. 1.
ADMISSIBLE METRICS IN THE σkYAMABE EQUATION
, 2008
"... In most previous works on the existence of solutions to the σkYamabe problem, one assumes that the initial metric g0 is kadmissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition. ..."
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Cited by 1 (0 self)
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In most previous works on the existence of solutions to the σkYamabe problem, one assumes that the initial metric g0 is kadmissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition.
Blowingup solutions for the Yamabe equation
, 2014
"... Let ðM; gÞ be a smooth, compact Riemannian manifold of dimension Nb 3. We consider the almost critical problem ðPeÞ Dguþ N 2 4ðN 1Þ Scalg u u ðNþ2Þ=ðN2Þþe in M; u> 0 in M; where Dg denotes the LaplaceBeltrami operator, Scalg is the scalar curvature of g and e a R is a small parameter. It is ..."
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Let ðM; gÞ be a smooth, compact Riemannian manifold of dimension Nb 3. We consider the almost critical problem ðPeÞ Dguþ N 2 4ðN 1Þ Scalg u u ðNþ2Þ=ðN2Þþe in M; u> 0 in M; where Dg denotes the LaplaceBeltrami operator, Scalg is the scalar curvature of g and e a R is a small parameter. It is known that problem ðPeÞ does not have any blowingup solutions when e % 0, at least for Na 24 or in the locally conformally flat case, and this is not true anymore when e & 0. Indeed, we prove that, if Nb 7 and the manifold is not locally conformally flat, then problem ðPeÞ does have a family of solutions which blowup at a maximum point of the function x! jWeylgðxÞjg as e & 0: Here Weylg denotes the Weyl curvature tensor of g:
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
F.: Blowup phenomena for the Yamabe equation II
 J. Differential Geom
, 2009
"... Let (M,g) be a compact Riemannian manifold of dimension n ≥ 3. The Yamabe problem is concerned with finding metrics of constant scalar curvature in the conformal class of g. This problem leads to a semilinear elliptic PDE for the conformal factor. More precisely, a conformal metric of the ..."
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Cited by 5 (1 self)
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Let (M,g) be a compact Riemannian manifold of dimension n ≥ 3. The Yamabe problem is concerned with finding metrics of constant scalar curvature in the conformal class of g. This problem leads to a semilinear elliptic PDE for the conformal factor. More precisely, a conformal metric of the
Blowup solutions for linear perturbations of the Yamabe equation. Concentration Analysis and Applications to Pde: Icts Workshop
, 2012
"... Abstract. For a smooth, compact Riemannian manifold (M, g) of dimension N ≥ 3, we are interested in the critical equation ..."
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Cited by 3 (2 self)
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Abstract. For a smooth, compact Riemannian manifold (M, g) of dimension N ≥ 3, we are interested in the critical equation
THE SECOND YAMABE INVARIANT
, 2008
"... Abstract. Let (M, g) be a compact Riemannian manifold of dimension n≥3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of th ..."
Results 1  10
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