Existence of positive solutions to a class of p(x)p(x)-Laplacian equations with singular nonlinearities

This paper investigates the following p(x)p(x)-Laplacian equations {−Δp(x)u≔−div(|∇u|p(x)−2∇u)=λK(x)f(x,u)+βuq(x),in Ω,u=0,on ∂Ω, where −Δp(x)−Δp(x) is called p(x)p(x)-Laplacian, f(x,⋅)f(x,⋅) is decreasing, and ff is singular, i.e., f(x,s)→+∞f(x,s)→+∞ as s→0+s→0+ for each x∈Ω¯. The existence of a positive solution is given. Especially, we do not restrict the growth speed of f(x,u)f(x,u) tends to +∞+∞ as u→0+u→0+.