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

We study the second order nonlinear differential equation \begin{equation*} u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j} b_{j}(x)k_{j}(u) = 0, \end{equation*} where $\alpha_{i},\beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0,L\mathclose{]}$, 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 $\beta_{j}$ 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.