Liouville-type theorem for an indefinite logarithmic Laplacian equation

In this paper, we consider fractional equations involving the logarithmic Laplacian with indefinite nonlinearities: [Formula: see text] where [Formula: see text] represents a Lipschitz coercive epigraph. Our investigation begins by establishing a boundary estimate for antisymmetric functions and demonstrating the monotonicity of bounded positive solutions in coercive epigraphs, via the direct method of moving planes. Subsequently, we present a Liouville-type theorem for this nonlocal problem.