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.
Categories: math.CA