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Liouville theorems for nonnegative solutions to Hardy-Hénon type system on a half space. (English) Zbl 1480.35078

Summary: In this paper, we are concerned with the Hardy-Hénon type system on a half space \(\mathbb{R}^n_+\) \[ \begin{cases} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,v), \quad u(x)\geq 0, \quad x\in \mathbb{R}^n_+, \\ (-\Delta)^{\frac{\beta}{2}}v(x)=g(x,u), \quad v(x)\geq 0, \quad x\in \mathbb{R}^n_+ \end{cases} \] with Dirichlet boundary conditions, where \(n\geq 1\), \(n>\max \{\alpha ,\beta\}\) and \(0<\alpha,\beta \leq 2\). We derive Liouville theorems (i.e., the non-existence of nontrivial nonnegative solutions) provided that \(f\) and \(g\) satisfy certain subcritical growth conditions (see Theorem 1.6). The argument used in our proof is the method of scaling spheres developed in [the first author and Qin, Guolin, “Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres”, Preprint, arXiv:1810.02752]. Our results generalize the Liouville theorems for single Lane-Emden equation in [W. Chen et al., Adv. Math. 274, 167–198 (2015; Zbl 1372.35332)] and Hardy-Hénon-type equation in [Dai and Qin, loc. cit.] to system and the Liouville theorems for system of Lane-Emden equations (\(f=v^p\), \(g=u^q\)) in [the first author et al., Potential Anal. 46, No. 3, 569–588 (2017; Zbl 1364.35421)] to system of equations with general Hardy-Hénon-type nonlinearities \(f(x, v)\) and \(g(x, u)\).

MSC:

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J57 Boundary value problems for second-order elliptic systems
35J61 Semilinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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