Monotonicity of positive solutions for an indefinite logarithmic Laplacian equation

In this paper, we investigate a nonlocal equation involving the logarithmic Laplacian with indefinite nonlinearities: \begin{equation*} \left{ \begin{array}{ll} L_Δu(x)=a(x_n)f(u), & x\inΩ, \ u(x)=0,& x\in $\mathbb{R}$^n\backslashΩ. \end{array} \right. \end{equation*} Here, $\mathrm{\Omega }$ represents a Lipschitz coercive epigraph. To achieve our objectives, we develop a boundary estimate for antisymmetric functions, enabling us to establish the monotonicity and nonexistence of bounded positive solutions for the above problem using the direct method of moving planes.