Maximum principles and moving planes method for the fractional $p(x,\cdot )$-Laplacian

In this paper, we investigate the monotonicity of solutions for a nonlinear equations involving the fractional Laplacian with variable exponent. We first prove different maximum principles involving this operator. Then we employ the direct moving planes method to obtain monotonicity of solutions to a nonlinear equations in which the fractional laplacian with variable exponent is present. Note that, there are no results studying the monotonicity of solutions for local or nonlocal equations with variables exponent. Our results are new in this setting and includes a self-contained techniques.