A multiplicity results for a singular equation involving the p(x)-Laplace operator

The purpose of this paper is to study the singular problem involving the p(x)-Laplace operator: (P-lambda) {-Delta(p(x))u = lambda/u(delta)(x) + f(x,u) in Omega, u > 0 in Omega, u = 0, on partial derivative Omega. where Omega subset of R-N, (N >= 2) be a bounded domain with C-2 boundary, lambda is a positive parameter and p(x), delta(x) and f(x, u) are assumed to satisfy the assumptions (H0)-(H4) in the Introduction. We employ variational techniques in order to show the existence of a number Lambda is an element of (0, infinity) such that problem (P-lambda) has two solutions for lambda is an element of (0, Lambda), one solution for lambda = Lambda and no solutions for lambda > Lambda . To obtain multiple (at least two distinct, positive) solutions of problem (P-lambda), we need to prove two new results: a regularity result for solutions to problem (P-lambda) in C-1,C-alpha((Omega) over bar) with some alpha is an element of (0, 1), and a strong comparison principle.