Existence of positive solutions to Dirichlet boundary-value problems for some recursive differential systems. (Q417023)

The author considers the nonlinear problemNEWLINENEWLINE\(u_j''+f_{j+1}(u_{j+1})=0\), \(j=1,\dots ,m-1\),NEWLINENEWLINE\(u_m''+f_1(u_1)=0\) in \((-R,R)\), \(u_j(\pm R)=0\), \(j=1,\dots ,m\),NEWLINENEWLINEwhere \(f_j\)'s are nonnegative, nondecreasing, continuous functions. It is shown that under certain conditions on \(f_j\)'s, which can be understood as superlinearity at \(\infty \) and at zero of the system, the problem has at least one positive solution which is symmetric with respect to the origin. A variant of this result is also offered. Further, the author considers the nonlinear elliptic systemNEWLINENEWLINE\(\Delta u_j+f_{j+1}(u_{j+1})=0\), \(j=1,\dots ,m-1\),NEWLINENEWLINE\(\Delta u_m+f_1(u_1)=0\) in \(\Omega (a,b)\), \(u_j=0\) on \(\partial \Omega (a,b)\), \(j=1,\dots ,m\). Conditions in terms of certain (power) bounds for \(f_j\)'s are given which guarantee the existence of at least one positive radial solution to the elliptic problem. The proofs consist of obtaining a priori estimates on the positive solutions, and then applying well-known properties of compact mappings taking a cone in a Banach space into itself.