Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification

  • Ting-Ying Chang

    School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia
  • Florica C. Cîrstea

    School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Abstract

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form

where denotes the open ball with radius centred at in . We assume that , and are positive functions associated with regularly varying functions of index and at and respectively, satisfying and . We prove that the condition is sharp for the removability of all singularities at for the positive solutions of , where denotes the “fundamental solution” of (the Dirac mass at 0) in , subject to . If , we show that any non-removable singularity at for a positive solution of (0.1) is either weak (i.e., ) or strong (). The main difficulty and novelty of this paper, for which we develop new techniques, come from the explicit asymptotic behaviour of the strong singularity solutions in the critical case, which had previously remained open even for . We also study the existence and uniqueness of the positive solution of with a prescribed admissible behaviour at and a Dirichlet condition on .

Cite this article

Ting-Ying Chang, Florica C. Cîrstea, Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 6, pp. 1483–1506

DOI 10.1016/J.ANIHPC.2016.12.001