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″+λx−2yσ=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 λγσ−1⩽Σ⋆. For 0<λγσ−1<Σ⋆, there are multiple positive solutions, while if γ=(λ−1Σ⋆)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 −Δu=d−2(x)uσ in Ω, where Ω⊂RN, N⩾2, is a smooth bounded domain and d(x)=dist(x,∂Ω).