Multiple positive solutions for a class of semipositone Neumann problems with singular φ-laplacian

We study the existence of multiple positive solutions for a Neumann problem with singular φ-Laplacian {-(φ(u′))′=λf(u),x∈(0,1),u′(0)=0=u′(1),where λ is a positive parameter, φ(s)=s1-s2,f∈C1([0,∞),ℝ),f′(u)>0foru>0, and for some 0<β<θ such that f(u)<0 for u∈[0,β) (semipositone) and f(u)>0 for u > β. Under some suitable assumptions, we obtain the existence of multiple positive solutions of the above problem by using the quadrature technique. Further, if f ∈ C2([0,β)∪(β,∞),ℝ), f″(u)≥0 for u ∈[0,β) and f″(u) ≤0 for u ∈(β,∞), then there exist exactly 2n+1 positive solutions for some interval of λ, which is dependent on n and θ. Moreover, We also give some examples to apply our results.