Electron. J. Diff. Equ., Vol. 2013 (2013), No. 33, pp. 1-10.

Positive blowup solutions for some fractional systems in bounded domains

Ramzi Alsaedi

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.

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Ramzi Alsaedi
Department of Mathematics, College of Science and Arts
King Abdulaziz University, Rabigh Campus, P. O. Box 344
Rabigh 21911, Saudi Arabia
email: ramzialsaedi@yahoo.co.uk

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