Positive solutions for dirichlet problems of singular quasilinear elliptic equations via variational methods

This paper studies the existence and multiplicity of positive solutions of the following problem: {-Δ_pu = u^β/ δγ(x) + λa(x)/u^α, in Ω, u > 0, in Ω, u = 0, on ∂Ω, (*) where Ω⊂ R^N(N ≥ 3) is a smooth bounded domain, Δ_pu = div(|Δu| ^p-2Δu), 1 < p < N, and 0 < α < 1, p - 1 < β < p* - 1 (p* = Np/(N - p)) and 0 < γ < N + ((β+ 1)(p - N)/p) are three constants. Also δ(x) = dist(x, ∂Ω)), a ∈ L^p and λ > 0 is a real parameter. By using the direct method of the calculus of variations, Ekeland's Variational Principle and an idea of G. Tarantello, it is proved that problem (*) has at least two positive weak solutions if A is small enough.