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

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 -div (A(vertical bar x vertical bar) vertical bar del vertical bar u vertical bar(p-2) del u) + b(x)h(u) = 0 in B1\{0), where B-r denotes the open ball with radius r > 0 centred at 0 in R-N (N >= 2). We assume that A is an element of C-1(0,1], b is an element of C((B-1) over bar\{0}) and h is an element of C[0, infinity) are positive functions associated with regularly varying functions of index upsilon, sigma and q at 0, 0 and infinity respectively, satisfying q > p-1 > 0 and upsilon-sigma < p < N + upsilon. We prove that the condition b(x)h(Phi) is not an element of L-1 (B-1/2) is sharp for the removability of all singularities at 0 for the positive solutions of (0.1), where (Phi) denotes the "fundamental solution" of -div (A(vertical bar x vertical bar) vertical bar del vertical bar u vertical bar(p-2) del u) = delta(0) (the Dirac mass at 0) in B-1, subject to Phi vertical bar partial derivative(B1) = 0. If b(x) h (Phi) is an element of L-1 (B-1/2), we show that any non-removable singularity at 0 for a positive solution of (0.1) is either weak (i.e., lim(vertical bar x vertical bar)-> 0 u (x)/<Phi(vertical bar x vertical bar) is an element of (0, infinity)) or strong (lim vertical bar x vertical bar -> 0u(x)/Phi(vertical bar x vertical bar) = infinity). 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 A = 1. We also study the existence and uniqueness of the positive solution of (0.1) with a prescribed admissible behaviour at 0 and a Dirichlet condition on partial derivative B-1. (C) 2016 Elsevier Masson SAS. All rights reserved.