Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 33, pp. 1-10.
Title: Positive blowup solutions for some fractional systems in bounded domains
Author: Ramzi Alsaedi (King Abdulaziz Univ., Rabigh, Saudi Arabia)
Abstract:
Using some potential theory tools and the Schauder fixed point theorem,
we prove the existence of a positive continuous weak solution
for the fractional system
$$
( -\Delta )^{\alpha/2}u+ p(x)u^{\sigma }v^{r}=0,\quad
(-\Delta)^{\alpha/2}v+q(x)u^{s}v^{\beta }=0
$$
in a bounded $ C^{1,1}$-domain D in $\mathbb{R}^{n}$ $(n\geq 3)$,
subject to Dirichlet conditions, where $0<\alpha <2$,
$\sigma ,\beta \geq 1$, $s,r\geq 0$. The potential functions p,q
are nonnegative and required to satisfy some adequate hypotheses related
to the Kato class $K_{\alpha }(D)$. We also investigate the global
behavior of such solution.
Submitted November 2, 2012. Published January 30, 2013.
Math Subject Classifications: 26A33, 34B27, 35B44, 35B09.
Key Words: Fractional nonlinear systems, Green function, positive solutions,
maximum principle.