Monotonicity of positive solutions to equations involving fractional p -Laplacian in coercive epigraph

Abstract In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional p -Laplacian: equation*cases (-) ₚ^ u=f (x, u, u), \ \ u 0, \ \ in\ \ E, \\\ \ \ \ \ \ u0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ \ R^n E, casesequation* where E R^n is a coercive epigraph, i. e. , there exists a continuous function: \, R^n-1 R satisfying equation*|ₗ'|+ (x') =+, equation* such that E: =\x= (x', x₍) R^{n|\, x₍ (x') \}, where x': = (x₁,. . . , x₍-₁) R^n-1. Under some mild assumptions on the nonlinearity f (x, u, u), we prove strict monotonicity of positive solutions to the above Dirichlet problems involving fractional p -Laplacian in coercive epigraph E.