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Electronic Journal of Differential Equations Electron. J. Diff. Equ., Vol. 2012 (2012), No. 208, pp. 1-11.

Existence of positive solutions for nonlinear fractional systems in bounded domains

Imed Bachar

We prove the existence of positive continuous solutions to the nonlinear fractional system
$$\displaylines{ (-\Delta|_D) ^{\alpha/2}u+\lambda g(.,v) =0,  \cr (-\Delta|_D) ^{\alpha/2}v+\mu f(.,u)  =0, }$$
in a bounded $C^{1,1}$-domain $D$ in $\mathbb{R}^n$ $(n\geq 3)$, subject to Dirichlet conditions, where $0<\alpha \leq 2$, $\lambda $ and $\mu $ are nonnegative parameters. The functions f and g are nonnegative continuous monotone with respect to the second variable and satisfying certain hypotheses related to the Kato class.

Submitted September 8, 2012. Published November 25, 2012.
Math Subject Classifications: 35J60, 34B27, 35B44.
Key Words: Fractional nonlinear systems; Green function; positive solutions; blow-up solutions

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Imed Bachar
King Saud University College of Science Mathematics
Department P.O. Box 2455
Riyadh 11451 Kingdom of Saudi Arabia

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