Electron. J. Differential Equations, Vol. 2017 (2017), No. 105, pp. 1-11.

Liouville-type theorems for an elliptic system involving fractional Laplacian operators with mixed order

Mohamed Jleli, Bessem Samet

Abstract:
We study the nonexistence of nontrivial solutions for the nonlinear elliptic system
$$\displaylines{ G_{\alpha,\beta,\theta}(u^{p},u^{q}) = v^{r}\cr G_{\lambda,\mu,\theta}(v^{s},v^{t}) = u^{m}\cr u,v\geq 0, }$$
where $0<\alpha,\beta,\lambda,\mu\leq 2$, $\theta\geq 0$, $m>q\geq p\geq 1$, $r>t\geq s\geq 1$, and $G_{\alpha,\beta,\theta}$ is the fractional operator of mixed orders $\alpha,\beta$, defined by
$$ G_{\alpha,\beta,\theta}(u,v)=(-\Delta_x)^{\alpha/2}u +|x|^{2\theta} (-\Delta_y)^{\beta/2}v, \quad \text{in }\mathbb{R}^{N_1} \times \mathbb{R}^{N_2}. $$
Here, $(-\Delta_x)^{\alpha/2}$, $0<\alpha<2$, is the fractional Laplacian operator of order $\alpha/2$ with respect to the variable $x\in \mathbb{R}^{N_1}$, and $(-\Delta_y)^{\beta/2}$, $0<\beta<2$, is the fractional Laplacian perator of order $\beta/2$ with respect to the variable $y\in \mathbb{R}^{N_2}$. Via a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters.

Submitted February 2, 2017. Published April 18, 2017.
Math Subject Classifications: 35B53, 35R11.
Key Words: Liouville-type theorem; nonexistence; fractional Grushin operator.

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Mohamed Jleli
Department of Mathematics
College of Science, King Saud University
P.O. Box 2455, Riyadh 11451, Saudi Arabia
email: jleli@ksu.edu.sa
Bessem Samet
Department of Mathematics
College of Science, King Saud University
P.O. Box 2455, Riyadh 11451, Saudi Arabia
email: bessem.samet@gmail.com

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