  We consider the fractional elliptic inequality with variable-exponent
nonlinearity $$ (-\Delta)^{\frac{\alpha}{2}} u+\lambda\, \Delta u \geq
|u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $\alpha\in (0,2)$,
$\lambda\in\mathbb{R}$ is a constant, $p: \mathbb{R}^N\to (1,\infty)$ is a
measurable function, and $(-\Delta)^{\frac{\alpha}{2}}$ is the fractional
Laplacian operator of order $\frac{\alpha}{2}$. A Liouville-type theorem is
established for the considered problem. Namely, we obtain sufficient conditions
under which the only weak solution is the trivial one. Next, we extend our
study to systems of fractional elliptic inequalities with variable-exponent
nonlinearities. Besides the consideration of variable-exponent nonlinearities,
the novelty of this work consists in investigating sign-changing solutions to
the considered problems. Namely, to the best of our knowledge, only
nonexistence results of positive solutions to fractional elliptic problems were
invetigated previously. Our approach is based on the nonlinear capacity method
combined with a pointwise estimate of the fractional Laplacian of some test
functions, which was derived by Fujiwara (2018) (see also Dao and Reissig
(2019)). Note that the standard nonlinear capacity method cannot be applied to
the considered problems due to the change of sign of solutions.
