Electron. J. Diff. Equ., Vol. 2016 (2016), No. 146, pp. 1-10.

Liouville-type theorems for elliptic inequalities with power nonlinearities involving variable exponents for a fractional Grushin operator

Mohamed Jleli, Mokhtar Kirane, Bessem Samet

Abstract:
We establish Liouville-type theorems for the elliptic inequality
$$
 u\geq 0,\quad G_{\alpha,\beta,\theta}\big(u^{p(x,y)},u^{q(x,y)}\big)
 \geq u^{r(x,y)}, \quad (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
 $$
where $G_{\alpha,\beta,\theta}$, $0<\alpha,\beta<2$, $\theta\geq 0$, is the fractional Grushin operator of mixed orders $\alpha,\beta$, defined by
$$
 G_{\alpha,\beta,\theta}(u,v)=(-\Delta_x)^{\alpha/2}u
 +|x|^{2\theta} (-\Delta_y)^{\beta/2}v,
 $$
where, $(-\Delta_x)^{\alpha/2}$ is the fractional Laplacian operator of order $\alpha$ with respect to the variable $x\in \mathbb{R}^{N_1}$, and $(-\Delta_y)^{\beta/2}$ is the fractional Laplacian operator of order $\beta$ with respect to the variable $y\in \mathbb{R}^{N_2}$. Here, $p,q,r: \mathbb{R}^{N_1}\times\mathbb{R}^{N_2}\to [1,\infty)$ are measurable functions satisfying certain conditions.

Submitted March 1, 2016. Published June 14, 2016.
Math Subject Classifications: 35B53, 35R11, 35J70.
Key Words: Liouville-type theorem; elliptic inequalities; variable exponent; 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
Mokhtar Kirane
LaSIE, Faculté des Sciences et Technologies
Université de La Rochelle
Avenue M. Crépeau, 17042 La Rochelle, France
email: mokhtar.kirane@univ-lr.fr
Bessem Samet
Department of Mathematics, College of Science
King Saud University, P.O. Box 2455
Riyadh 11451, Saudi Arabia
email: bsamet@ksu.edu.sa

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