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

Ting-Ying Chang, Florica C. Cîrstea

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. 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 p-Laplacian type equations. We also study the existence and uniqueness of the positive solution of our equation with a prescribed admissible behaviour at zero and a non-homogeneous Dirichlet boundary condition.

Keywords: divergence-form elliptic equations, isolated singularities, regular variation theory.

AMS Subject Classification: Primary 35J25; secondary 35B40, 35J60.