Electronic Journal of Differential Equations
In this article, we study the existence of positive solutions for the p(x)-Laplacian Dirichlet problem −∆p(x)u = λf(x, u) in a bounded domain Ω ⊂ RN. The singular nonlinearity term f is allowed to be either f(x, s) → +∞, or f(x, s) → +∞ as s → 0+ for each x ∈ Ω. Our main results generalize the results in  from constant exponents to variable exponents. In particular, we give the asymptotic behavior of solutions of a simpler equation which is useful for finding supersolutions of differential equations with variable exponents, which is of independent interest.
Liu, Jingjing, Qihu Zhang, Chunshan Zhao.
"Existence of Positive Solutions for p(x)-Laplacian Equations with a Singular Nonlinear Term."
Electronic Journal of Differential Equations, 2014 (155): 1-21.