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Regularity and existence of positive solutions for a fractional system

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The first author is partially supported by NSFC 11701207
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  • We consider the nonlinear fractional elliptic system

    $ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), & \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), & \text{in}\, \, \, \Omega, \\ u = v = 0, & \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $

    where $ 0<\alpha_1, \alpha_2<2 $ and $ \Omega $ is a bounded domain with $ C^2 $ boundary in $ \mathbb{R}^n $. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for $ 0<\alpha_1, \alpha_2<1 $ and $ 1<\alpha_1, \alpha_2 <2 $ respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.

    Mathematics Subject Classification: Primary: 35B09, 35S15; Secondary: 35B44, 35B45, 35B65.


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