Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

Abstract

We study the semilinear elliptic equation

Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms

“…The equality case in (1.4) naturally leads to a proper differential equation and has even a longer history. We only mention here the seminal work of Gidas and Spruck [19] for the semilinear case with power type nonlinearity but also some more recent results [10], [15], [23] dealing with other different situations. A systematic study of the inequality L A u = −div[A(x, u, ∇u)] ≥ |x| σ u q in Ω, along with the corresponding system…”

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

“…have been studied in detail in [9,17,40,41,46,47]. Yamabe type equations ∆u + p(x)u s + q(x)u = 0 are also of form (1.1) with a powerlike nonlinearity.…”

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

“…where p, q are real exponents, a, b are sufficiently smooth functions and Γ ∈ C 2 (R, R) (see [9,20,52,53,60,61] and the references therein for gradient estimates and related results in this direction). Another class of equations that have been extensively studied and whose nonlinearity is in the form of superposition of power-like nonlinearities are the Yamabe equations (see, e.g., [6,13,24,30]).…”