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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