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.

Show me the PDF file (240 KB), TEX file for this article.

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

Return to the EJDE web page