Positive solutions for systems of competitive fractional differential equations

Abstract

Using potential theory arguments, we study the existence and boundary behavior of positive solutions in the space of weighted continuous functions, for the fractional differential system $$\displaylines{ D^{\alpha }u(x)+p(x)u^{a_1}(x)v^{b_1}(x) =0\quad \text{in }(0,1),\quad \lim_{x\to 0^{+}}x^{1-\alpha }u(x)=\lambda >0, \cr D^{\beta }v(x)+q(x)v^{a_2}(x)u^{b_2}(x) = 0\quad \text{in }(0,1),\quad \lim_{x\to 0^{+}}x^{1-\beta }v(x)=\mu >0, }$$ where $\alpha,\beta \in (0,1)$, $a_i>1$, $b_i\geq 0$ for $i\in \{1,2\}$ and $p,q$ are positive continuous functions on $(0,1)$ satisfying a suitable condition relying on fractional potential properties.

Keywords

• Fractional differential system
• positive solution
• potential theory
• Schauder fixed point theorem