On a class of semilinear elliptic problems with singular nonlinearities

摘要

This paper is devoted to a class of semilinear elliptic problems (P)λ-Δu=λf(x)(1-u)β,x∈Ω,u=0,x∈∂Ω,with 0 < u(x) < 1 in a bounded domain Ω⊂RN (N ⩾ 1). Here λ and β are positive parameters and f(x)∈C(Ω¯) is a nonnegative function. By the sub–super solutions method we show that there exists a critical parameter λ∗=λ∗(Ω,β,f) such that (P)λ has at least one solution including a unique minimal solution for λ ∈ (0, λ∗) and no solution for λ > λ∗. Moreover we get the rigorous bounds on λ∗. We also discuss properties of minimal solutions of (P)λ including regularity and monotonicity.