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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 formwhere B r denotes the open ball with radius r > 0 centred at 0 in R N (N ≥ 2). We assume thatare positive functions associated with regularly varying functions of index ϑ, σ and q at 0, 0 and ∞ respectively, satisfying q > p − 1 > 0 and ϑ − σ < p < N + ϑ. We prove that the condition b(x) h(Φ) L 1 (B 1/2 ) is sharp for the removability of all singularities at 0 for the positive solutions of (0.1), where Φ denotes the "fundamental solution" of −div (A(|x|) |∇u| p−2 ∇u) = δ 0 (the Dirac mass at 0) in B 1 , subject to Φ| ∂B 1 = 0. If b(x) h(Φ) ∈ 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 |x|→0 u(x)/Φ(|x|) ∈ (0, ∞)) or strong (lim |x|→0 u(x)/Φ(|x|) = ∞). 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 ∂B 1 .

<abstract><p>In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. ^{[<xref ref-type="bibr" rid="b1">1</xref>]} by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 &gt; 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni ^{[<xref ref-type="bibr" rid="b2">2</xref>]} to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. ^{[<xref ref-type="bibr" rid="b1">1</xref>]}).</p></abstract>

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