Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities

We study the second order nonlinear differential equation u''+∑_i α_ia_i(x)g_i(u) − ∑j β_jb_j(x)k_j(u)=0, where α_i,β_j>0, a_i(x), b_j(x) are non-negative Lebesgue integrable functions defined in [0,L], and the nonlinearities g_i(s), k_j(s) are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation u''+a(x)u^p=0, with p>1. When the positive parameters β_j are sufficiently large, we prove the existence of at least 2^m-1 positive solutions for the Sturm-Liouville boundary value problems associated with the equation. The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets. Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.