Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

Abstract

We consider the fractional elliptic inequality with variable-exponent nonlinearity

$$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}} u+\lambda \, \Delta u \ge |u|^{p(x)}, \quad x\in {\mathbb {R}}^N, \end{aligned}$$

where $N\ge 1$, $\alpha \in (0,2)$, $\lambda \in {\mathbb {R}}$ is a constant, $p: {\mathbb {R}}^N\rightarrow (1,\infty )$ 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 investigated 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 (Math Methods Appl Sci 41:4955–4966, 2018) [see also Dao and Reissig (A blow-up result for semi-linear structurally damped $\sigma$-evolution equations, arXiv:1909.01181v1, 2019)]. Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.