We study existence, multiplicity, and the behavior, with respect to λ, of positive radially symmetric solutions of −Δu=λh(x,u) in annular domains in double-struckRN,N≥2. The nonlinear term has a superlinear local growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive and concave function a. For this, we combine several methods such as the sub and supersolutions method, a priori estimates and degree theory.