Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities

We study the existence of multiple positive solutions of -Δ u = λ u-q +uᵖ in Ω with homogeneous Dirichlet boundary condition, where Ω is a bounded domain in RN, λ >0, and 0 < q ≤ 1 < p ≤ (N+2)/(N-2). We show by a variational method that if λ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of $q=1$ or $p=(N+2)/(N-2)$. We also study the regularity of positive weak solutions.