Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity


Authors: Steven D. Taliaferro and Lei Zhang
Journal: Proc. Amer. Math. Soc. 131 (2003), 2895-2902
MSC (2000): Primary 35J60, 53C21
Published electronically: January 28, 2003
MathSciNet review: 1974347
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the conformal scalar curvature problem

\begin{displaymath}k(x) u^{\frac{n+2}{n-2}} \le -\Delta u \le u^{\frac{n+2}{n-2}}\qquad \hbox {in} \qquad \mathbf{R}^{n}, n\ge 3,\end{displaymath}

where $k : \mathbf{R}^{n} \to (0,1]$ 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 $\infty $ is that $k$ be less than 1 on a sequence of points in $\mathbf{R}^{n}$ which tends to $\infty $.


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

  • 1. L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297. MR 90c:35075
  • 2. C.-C. Chen and C.-S. Lin, Estimates of the scalar curvature equation via the method of moving planes, Comm. Pure Appl. Math. 50 (1997), 971-1017. MR 98k:35051
  • 3. N. Korevaar, R. Mazzeo, F. Pacard, and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math. 135 (1999), 233-272. MR 2001a:35055
  • 4. M.-C. Leung, Blow-up solutions of nonlinear elliptic equations in $\mathbf{R}^{n}$ with critical exponent, preprint.
  • 5. C.-S. Lin, Estimates of the scalar curvature equation via the method of moving planes III, Comm. Pure Appl. Math. 53 (2000), 611-646. MR 2001i:53056
  • 6. S. D. Taliaferro, On the growth of superharmonic functions near an isolated singularity I, J. Differential Equations 158 (1999), 28-47. MR 2000j:35288
  • 7. S. D. Taliaferro, Isolated singularities of nonlinear elliptic inequalities, Indiana Univ. Math. J. 50 (2001), 1885-1897.
  • 8. S. D. Taliaferro, Local behavior and global existence of positive solutions of $au^{\lambda }\le -\Delta u \le u^{\lambda }$, Ann. Inst. H. Poincare Anal. Non Lineaire, in press, see http://www.math.tamu.edu/$\sim $steven.taliaferro/selpubs.html.
  • 9. Lei Zhang, Refined asymptotic estimates for conformal scalar curvature equation via moving sphere method, J. Functional Analysis, in press.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J60, 53C21

Retrieve articles in all journals with MSC (2000): 35J60, 53C21


Additional Information

Steven D. Taliaferro
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: stalia@math.tamu.edu

Lei Zhang
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: lzhang@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06932-6
PII: S 0002-9939(03)06932-6
Received by editor(s): March 1, 2002
Received by editor(s) in revised form: April 11, 2002
Published electronically: January 28, 2003
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society