Existence of positive solutions for nonlinear systems

Abstract

This paper deals with the existence of positive solutions for the nonlinear system q(t)\phi(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $\textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)\hbox{\rm :} [0,1]\to (0,\oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)\phi(p(t)u'_{i}(t)))'+\lambda h_{i}(t)g^{i} (\textbf{u})=0, \quad 0<t<1,\quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.