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

In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional p -Laplacian: {equation*}{cases}(-Δ)_p^α u=f(x,u,∇ u),\ \ u > 0,\ \ in\ \ E,\\\ \ \ \ \ \ u≡0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ \ Rⁿ E,{cases}{equation*} where E ⊆ Rⁿ is a coercive epigraph, i.e., there exists a continuous function φ: \, Rⁿ⁻¹ → R satisfying {equation*}lim|x'|→+∞φ(x')=+∞,{equation*} such that E:==(x',xₙ) ∈ Rⁿ|\,xₙ > φ(x')\ , where x':= (x₁,...,xₙ₋₁) ∈ Rⁿ⁻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 .