The asymptotic behaviour of the unique solution for the singular Lane–Emden–Fowler equation

Abstract

By Karamata regular variation theory and constructing comparison functions, we show the exact asymptotic behaviour of the unique classical solution u ∈ C 2 (Ω) ∩ C(Ω) near the boundary to a singular Dirichlet problem −∆u = k(x)g(u), u > 0, x ∈ Ω, u| ∂Ω = 0, where Ω is a bounded domain with smooth boundary in R N ; g ∈ C 1 ((0, ∞), (0, ∞)), lim t→0 + g(ξ t) g(t) = ξ −γ , for each ξ > 0, for some γ > 0; and k ∈ C α loc (Ω) for some α ∈ (0, 1), is nonnegative on Ω, which is also singular near the boundary.  2005 Elsevier Inc. All rights reserved.

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

“…(1.2) Problem (1.2) was discussed in a number of works; see, for instance, [3,4,7,8,[13][14][15][20][21][22][23][24]26,27]. For b ≡ 1 on Ω: when g satisfies (g 1 ), Fulks and Maybee [7], Stuart [21], Crandall, Rabinowitz and Tartar [4] showed that problem (1.2) has a unique solution u ∈ C 2+α (Ω) ∩ C(Ω).…”

The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem

“…For λ = 0, problem (1.1) was discussed in a number of works, see, for instance, [5,7,13,15,22,26,28]. For b ≡ 1 on Ω, Crandall, Rabinowitz and Tartar [5,Theorems 2.2 and 2.7] showed that if g satisfies (g 1 ) then problem (1.1) has a unique solution u ∈ C 2+α (Ω) ∩ C(Ω).…”

“…In [28], by Karamata regular varying theory, a perturbed argument and constructing comparison functions, the author generalized the results in [9] with λ = 0 to the more general g and b, where b can be singular on the boundary.…”

Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation

“…The analogous problems for the Laplace operator were investigated by several authors. We refer the reader to the papers [3][4][5][6][7][8][9][10][11]14,18,19,[21][22][23] and references therein.…”