Superlinear ordinary elliptic systems involving parameters

Abstract

We use a unified approach to study certain classes of elliptic problems. More precisely, we consider a Dirichlet problem for a system of two ordinary differential equations which depends on two numerical parameters a and b, and with nonlinearities satisfying very general superlinear local growth conditions. Using the upper–lower solutions method, fixed point theorems of cone expansion/compression type and some degree–theoretic arguments, we prove that there exists a non–increasing function Γ of the parameter a such that the problem has (i) at least one positive solution for 0 ≤ b ≤ Γ(a), (ii) no positive solution for b> Γ(a), and (iii) at least two positive solutions for 0 <b< Γ(a). We apply the main results to a class of semilinear elliptic systems in both bounded annular domains and exterior domains with non–homogeneous Dirichlet boundary conditions. In addition, we apply our results to fourth–order boundary value problems. The nonlinearities may have singularities, as well as may vanish in parts of the domain.