We consider the nonlinear elliptic equation driven by the p-Laplacian and with a Carathéodory reaction depending on a parameter λ > 0, looking for positive solutions. We prove two bifurcation-type results. In the first the bifurcation occurs near 0 (for small values of λ > 0) and our setting incorporates problems with the combined effects of concave and convex nonlinearities. In the second, the bifurcation occurs near +∞ (for large values of λ > 0) and our setting incorporates as a special case p-logistic equations with superdiffusive reaction. Our approach is variational, based on minimax methods and truncation techniques.