In this paper we study the existence of positive smooth solutions for a class of singular (p(x), q(x))-Laplacian systems by using sub and supersolution methods.2010 Mathematics Subject Classification. 35J75; 35J48; 35J92.

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The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauder's fixed point theorem.2010 Mathematics Subject Classification. 35J75; 35J48; 35J92.

In this paper we establish existence of smooth positive solutions for a singular quasilinear elliptic system involving gradient terms. The approach combines sub-supersolutions method and Schauder's fixed point theorem.2010 Mathematics Subject Classification. 35J75; 35J48; 35J92.

We investigate the following quasilinear and singular system,            −∆ p1 u = u α1 v β1 in Ω −∆ p2 v = u α2 v β2 in Ω u > 0, v > 0, u, v = 0 on ∂Ω, (P) where Ω is an open bounded domain with smooth boundary, 1 < p i < ∞ and α i + β i < 0 for any i = 1, 2. We employ monotone methods in order to show the existence of a unique (positive) solution of problem (P) in some cone. When α i + β i > −1 for i = 1, 2, we prove a regularity result for solutions to problem (P) in C 1,β (Ω) with some β ∈ (0, 1). Furthermore, we show that min i=1,2 α i + β i > −1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C 1 (Ω).