On critical fractional systems with Hardy-Littlewood-Sobolev nonlinearities. (English) Zbl 1471.35005

Hong, Qianyu; Yang, Yang

Rocky Mt. J. Math. 50, No. 5, 1661-1683 (2020).

The authors study the following critical fractional Laplacian system with Choquard type nonlinearities: \[ \left\{ \begin{array} [c]{lll} (-\Delta)^s u=k_1\displaystyle\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{2_\mu^*}}{|x-y|^\mu}\right)|u|^{2_\mu^*-2}u+\frac{\alpha\gamma}{^{2_\mu^*}}\left(\int_{\mathbb{R}^N}\frac{|v(y)|^{\beta}}{|x-y|^\mu}\right)|u|^{\alpha-2}u & \mathrm{in} & \mathbb{R}^N,\\ (-\Delta)^s v=k_2\displaystyle\left(\int_{\mathbb{R}^N}\frac{|v(y)|^{2_\mu^*}}{|x-y|^\mu}\right)|v|^{2_\mu^*-2}v+\frac{\beta\gamma}{^{2_\mu^*}}\left(\int_{\mathbb{R}^N}\frac{|u(y)|^{\alpha}}{|x-y|^\mu}\right)|v|^{\beta-2}v & \mathrm{in} & \mathbb{R}^N, \end{array} \right. \] where \(s\in(0,1)\), \(N>2s\), \(k_1,k_2>0\), \(\gamma\neq 0\), \(\mu\in(0,N)\), \(\alpha,\beta>1\) and \(2_\mu^*=(2N-\mu)/(N-2s)\) is the fractional upper critical exponent in the Hardy-Littlewood-Sobolev inequality. By using minimization techniques over the Nehari set they provide existence and non-existence of solutions, for certain regions of parameters \((\alpha,\beta)\).

Reviewer: Kaye Silva (Goiânia)

MSC:

35A15	Variational methods applied to PDEs
35R09	Integro-partial differential equations
35R11	Fractional partial differential equations
35B33	Critical exponents in context of PDEs
35J47	Second-order elliptic systems
35J62	Quasilinear elliptic equations

Keywords:

Nehari manifold; fractional Laplacian; Choquard equation; Hardy-Littlewood-Sobolev critical exponent; fractional upper critical exponent

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