On nonnegative solutions of singular boundary-value problems for Emden-Fowler-type differential systems

Abstract

We investigate some boundary-value problems for an Emden--Fowler-type differential system \[ u_{1}^{\prime}=g_{1}(t)u_{2}^{\lambda_{1}},\;\;\;u_{2}^{\prime}=g_{2} (t)u_{1}^{\lambda_{2}} \] on a finite or infinite interval $I=[a,b)$, where $g_{i}:I\rightarrow \lbrack0,\infty)$ $(i=1,2)$ are locally integrable functions. We give the optimal, in a certain sense, sufficient conditions that guarantee the existence of a unique (at least of one) nonnegative solution, satisfying one of the two following boundary conditions: \[ \mathrm{i)}\ u_{1}(a)=c_{0},\;\;\;\lim_{t\rightarrow b}u_{1}(t)=c;\;\; \mathrm{ii)}\ u_{2}(a)=c_{0},\;\;\;\lim_{t\rightarrow b}u_{1}(t)=c, \] in case $0\leq c_{0} < c < +\infty$ (in case $c_{0}\geq0$, $c=+\infty$ and $\lambda_{1}\lambda_{2}>1$). Moreover, the global two-sided estimations of the above-mentioned solutions are obtained together with applications to differential equations with $p$-Laplacian.