Monotonicity of positive solutions for an indefinite logarithmic Laplacian equation

Abstract

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, $\Omega$ 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.

24 pages, 9 figures