The second order expansion of solutions to a singular Dirichlet boundary value problem

In this paper, we mainly study the second order expansion of classical solutions in a neighborhood of ∂Ω to the singular Dirichlet problem −Δu=b(x)g(u)+λa(x)f(u)−Δu=b(x)g(u)+λa(x)f(u), u>0u>0, x∈Ωx∈Ω, u|∂Ω=0u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RNRN, λ≥0λ≥0. The weight functions b,a∈Clocα(Ω) are positive in Ω and both may be vanishing or be singular on the boundary. The function g∈C1((0,∞),(0,∞))g∈C1((0,∞),(0,∞)) satisfies limt→0+⁡g(t)=∞limt→0+⁡g(t)=∞, and f∈C([0,∞),[0,∞))f∈C([0,∞),[0,∞)). We show that the nonlinear term λa(x)f(u)λa(x)f(u) does not affect the second order expansion of solutions in a neighborhood of ∂Ω to the problem for some kinds of functions b and a.