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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)
Published electronically: June 24, 2013
Keywords:Nonlinear elliptic equations, isolated singularities, regular variation theory, inverse square potentials

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Table of Contents


  • 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


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.

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