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Exact multiplicity result for the perturbed scalar curvature problem in $ \mathbb{R}^N $ $ (N \geq 3)$

Author: S. Prashanth
Journal: Proc. Amer. Math. Soc. 135 (2007), 201-209
MSC (2000): Primary 35B32, 35B33
Published electronically: June 28, 2006
MathSciNet review: 2280188
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Abstract: Let $ D^{1,2} (\mathbb{R}^N)$ denote the closure of $ C_0^\infty (\mathbb{R}^N)$ in the norm $ \Vert u\Vert _{D^{1,2} (\mathbb{R}^N)}^2 = \int\limits_{\mathbb{R}^N} \vert\nabla u\vert^2.$ Let $ N \geq 3$ and define the constants $ \alpha_N = N (N-2)$ and $ C_N = (N (N-2))^{\frac{N-2}{4}}.$ Let $ K \in C^2 (\mathbb{R}^N).$ We consider the following problem for $ \varepsilon \geq 0:$

$\displaystyle (P_\varepsilon)\qquad\qquad\quad \left\{\begin{array}{llll} \mbox... ...array}\right\} \mbox{ in } \mathbb{R}^N. \end{array} \right. \qquad\quad\qquad $

We show an exact multiplicity result for $ (P_\varepsilon)$ for all small $ \varepsilon >0$.

References [Enhancements On Off] (What's this?)

  • 1. A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of $ \Delta u + u^{\frac{N+2}{N-2}} =0,$ the scalar curvature problem in $ \mathbb{R}^N$ and related topics, J. Funct. Anal. 165 (1999), 117-149. MR 1696454 (2000d:35047)
  • 2. T. Aubin, Some nonlinear problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin (1998). MR 1636569 (99i:58001)
  • 3. G. Bianchi and H. Egnell, A note on Sobolev inequality, J. Funct. Anal. 100 (1991), 10-24. MR 1124290 (92i:46033)
  • 4. J. Kazdan and F. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and Scalar curvature, Ann. of Math. 101 (1975), 317-331. MR 0375153 (51:11349)

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Additional Information

S. Prashanth
Affiliation: TIFR Centre, Indian Institute of Science Campus, P.B. No. 1234, Bangalore - 560 012, India

Keywords: Yamabe problem, exact multiplicity, scalar curvature
Received by editor(s): April 14, 2005
Received by editor(s) in revised form: July 30, 2005
Published electronically: June 28, 2006
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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