We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u + f (x, u) = 0. We allow x → f (x, s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f (x, s)/s is below λ1 as s → 0 + and above λ1 as s → +∞. In particular, we can deal with the situation in which f (x, s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u + a(x)g(u) = 0, where we prove the existence of 2 n − 1 positive solutions when a(x) has n positive humps and a − (x) is sufficiently large.

We study the periodic boundary value problem associated with the second order nonlinear differential equationwhere g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c ∈ R and µ > 0 is a real parameter. We prove the existence of 2 m −1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and µ is sufficiently large. The proof is based on the extension of Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equationwhere g : [0, +∞[ → [0, +∞[ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when T 0 a(t) dt < 0 and λ > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations. * Work performed under the auspices of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

We study the periodic boundary value problem associated with the second order nonlinear equationwhere g(u) has superlinear growth at zero and sublinear growth at infinity. For λ, µ positive and large, we prove the existence of 3 m − 1 positive T -periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T -periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.