Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 58, pp. 1-9.
Title: Existence of bounded positive solutions of a nonlinear
differential system
Author: Sabrine Gontara (Fac. des sciences de Tunis, Tunisia)
Abstract:
In this article, we study the existence and nonexistence of solutions for the
system
$$\displaylines{
\frac{1}{A}(Au')'=pu^{\alpha }v^{s}\quad \hbox{on }(0,\infty ), \cr
\frac{1}{B}(Bu')'=qu^{r}v^{\beta }\quad \hbox{on }(0,\infty ), \cr
Au'(0)=0,\quad u(\infty )=a>0, \cr
Bv'(0)=0,\quad v(\infty )=b>0,
}$$
where $\alpha ,\beta \geq 1$, $s,r\geq 0$, p,q are two nonnegative
functions on $(0,\infty )$ and A, B satisfy appropriate conditions.
Using potential theory tools, we show the existence of a positive
continuous solution. This allows us to prove the existence of entire
positive radial solutions for some elliptic systems.
Submitted January 20, 2012. Published April 10, 2012.
Math Subject Classifications: 35J56, 31B10, 34B16, 34B27.
Key Words: Nonlinear equation; Green's function; asymptotic behavior;
singular operator; positive solution.