We consider one-dimensional (p,q)-Laplace problems:{−(φ(u′))′=λh(t)f(u),t∈(0,1),u(0)=0=au′(1)+g(λ,u(1))u(1), where λ>0, a≥0, φ(s):=|s|p−2s+|s|q−2s, 1<p<q<∞, h∈C((0,1),(0,∞)), f∈C((0,∞),R) with lims→0+f(s)∈(−∞,0)∪{−∞}, and g∈C((0,∞)×[0,∞),(0,∞)) such that g(r,s)s is nondecreasing with respect to s∈[0,∞). Classifying the behaviors of f near infinity, we establish the existence, multiplicity and nonexistence of positive solutions. In particular, we provide a sufficient condition on f to obtain a multiplicity result for the case when lims→∞f(s)sr−1∈(0,∞), 1<r<q, which is new even in semilinear problems (p=q=2). The proofs are based on a Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.