How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials


About this Title

Florica C. Cîrstea

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 227, Number 1068
ISBNs: 978-0-8218-9022-6 (print); 978-1-4704-1429-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1068
Published electronically: June 24, 2013
Keywords:Nonlinear elliptic equations, isolated singularities, regular variation theory, inverse square potentials

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Main results
  • Chapter 3. Radial solutions in the power case
  • Chapter 4. Basic ingredients
  • Chapter 5. The analysis for the subcritical parameter
  • Chapter 6. The analysis for the critical parameter
  • Chapter 7. Illustration of our results
  • Appendix A. Regular variation theory and related results

Abstract


In this paper, we consider semilinear elliptic equations of the form

where is a parameter with and is an open subset in with such that . Here, is a positive continuous function on which behaves near the origin as a regularly varying function at zero with index greater than . The nonlinearity is assumed continuous on and positive on with such that is bounded for small . We completely classify the behaviour near zero of all positive solutions of [[eqref]]one when is regularly varying at with index greater than (that is, for every ). In particular, our results apply to [[eqref]]one with as and as , where and are any real numbers. We reveal that the solutions of [[eqref]]one generate a very complicated dynamics near the origin, depending on the interplay between , , and , on the one hand, and the position of with respect to and , on the other hand. Our main results for appear here for the first time, as well as for the case . We establish a trichotomy of positive solutions of [[eqref]]one under optimal conditions, hence generalizing and improving through a different approach a previous result with Chaudhuri on [[eqref]]one with and . Moreover, recent results of the author with Du on (0.1) with are here sharpened and extended to any . In addition, we unveil a new single-type behaviour of the positive solutions of [[eqref]]one specific to . We also provide necessary and sufficient conditions for the existence of positive solutions of (0.1) that are comparable with the fundamental solutions of

In particular, for and , we find a sharp condition on such that the origin is a removable singularity for all non-negative solutions of [[eqref]]one, thus addressing an open question of Vázquez and Véron.

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

American Mathematical Society