Differential and Integral Equations

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

Mariella Cecchi, Zuzana Došlá, Ivan Kiguradze, and Mauro Marini

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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.

Article information

Differential Integral Equations, Volume 20, Number 10 (2007), 1081-1106.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B16 34B18 34B40 34C11


Cecchi, Mariella; Došlá, Zuzana; Kiguradze, Ivan; Marini, Mauro. On nonnegative solutions of singular boundary-value problems for Emden-Fowler-type differential systems. Differential Integral Equations 20 (2007), no. 10, 1081--1106. https://projecteuclid.org/euclid.die/1356039297

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