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

A multiplicity result for the singular ordinary differential equation y ″ + λ x −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 λ γ σ − 1 ⩽ Σ ⋆ . For 0 < λ γ 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 − Δ 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 ) .