Multiple positive solutions to a singular boundary value problem for a superlinear Emden–Fowler equation

Abstract

A multiplicity result for the singular ordinary differential equation y + λx −2 y σ = 0, posed in the interval (0, 1), with the boundary conditions y(0) = 0 and y(1) = γ , where σ > 1, λ > 0 and γ 0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ ∈ (0, σ/2] such that a solution to the above problem is possible if and only if λγ σ −1 Σ . For 0 < λγ σ −1 < Σ , there are multiple positive solutions, while if γ = (λ −1 Σ ) 1/(σ −1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x → 0 + is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem − u = d(x) −2 u σ in Ω, where Ω ⊂ R N , N 2, is a smooth bounded domain and d(x) = dist(x, ∂Ω).

Numerical treatment of nonlinear Emden–Fowler equation using stochastic technique

“…Step (8) (Termination Criteria) For all m q sub-population repeat steps (3)-(6) until the termination criteria meets.…”

“…Recently Wazwaz [7] applied Adomian decomposition method (ADM) to solve time dependent Emden-Fowler type of equations. The effect of competing parameters on the existence and non existence of positive solutions of singular Emden-Fowler equation is presented by Guedda [8]. Oscillation criteria for a class of second order Emden-Fowler delay dynamic equations on time scale has been solved by mean of Riccati transformation technique [9].…”