Existence of positive solutions to a class of -Laplacian equations with singular nonlinearities

This paper investigates the following p ( x ) -Laplacian equations { − Δ p ( x ) u ≔ − div ( | ∇ u | p ( x ) − 2 ∇ u ) = λ K ( x ) f ( x , u ) + β u q ( x ) , in  Ω , u = 0 , on  ∂ Ω , where − Δ p ( x ) is called p ( x ) -Laplacian, f ( x , ⋅ ) is decreasing, and f is singular, i.e., f ( x , s ) → + ∞ as 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 ) tends to + ∞ as u → 0 + .