Mathematics

We study the second order nonlinear differential equation \begindocument $ u'' + \sum\limits_i = 1^m α_ia_i(x)g_i(u) - \sum\limits_j = 1^m + 1 β_jb_j(x)k_j(u) = 0,\rm $ \enddocumentwhere $\alpha_i, \beta_j>0$, $a_i(x), b_j(x)$ 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.