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Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation

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The work was supported by National Natural Science Foundation of China (11871096 and 11671308)

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  • In this paper, we consider equations involving the fully nonlinear fractional order operator with homogeneous Dirichlet condition:

    $ \begin{cases} F_\alpha(u)(x) = f(x,u,\nabla u) \ \mbox{in} \ \Omega,\\ u>0, \ \mbox{in}\ \Omega; \ u\equiv0, \ \mbox{in}\ \mathbb R^n\backslash\Omega, \end{cases} $

    where $ \Omega $ is a domain(bounded or unbounded) in $ \mathbb R^n $ which is convex in $ x_1- $direction. By using some ideas of maximum principle, we prove that the solution is strictly increasing in $ x_1- $direction in the left half of $ \Omega $. Symmetry of solution is also proved. Meanwhile we obtain a Liouville type theorem on the half space $ \mathbb R^n_+ $.

    Mathematics Subject Classification: Primary: 35R11; Secondary: 35B33.


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