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Symmetry and nonexistence results for a fractional Hénon-Hardy system on a half-space. (English) Zbl 1421.35083

Summary: We study the fractional Henon-Hardy system \[\begin{aligned}\begin{cases}(-\Delta )^{s/2} u(x) = |x|^\alpha v^p(x), & x\in \mathbb{R}^n_+, \\(-\Delta )^{s/2} v(x) = |x|^\beta u^q(x), & x\in \mathbb{R}^n_+, \\ u(x)=v(x)=0, & x\in \mathbb{R}^n\setminus \mathbb{R}^n_+,\end{cases}\end{aligned}\] where \(n\ge 2\), \(0< s< 2\), \(\alpha,\beta >-s\) and \(p,q\ge 1\). We also consider an equivalent integral system. By using a direct method of moving planes, we prove some symmetry and nonexistence results for positive solutions under various assumptions on \(\alpha\), \(\beta\), \(p\) and \(q\).

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35R11 Fractional partial differential equations
35B06 Symmetries, invariants, etc. in context of PDEs
35B09 Positive solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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