Let Ω be a bounded domain in ^N (N 2) with smooth boundary ∂Ω. Let p(·) → be a continuous function such that p(0) = 0 and lim_{t → ± ∞}(p(t)/t) = p_± exist. We discuss the existence of nontrivial solutions of the Dirichlet problem, −Δu = p(u) in Ω, u = 0 on ∂Ω. Our hypotheses relate the position of the point (p₋, p₊) in the plane with the extended spectrum of −Δ. They seem to allow noticeably greater freedom for the numbers p_± than is usually found in the literature in the sense that the interval with end points p_± is here permitted to contain in its interior an eigenvalue of arbitrarily multiplicity of −Δ. It also appears that we significantly weaken some technical conditions on p(·) of another paper which treats the crossing of a simple eigenvalue.