Existence of positive solutions for p(x)-Laplacian problems

Abstract

We consider the system of differential equations -Δp(x)u = λ[g(x)α(u) + ƒ(v)] in Ω -Δq(x)v = λ[g(x)b(v) + h(u)] in Ω u = v = 0 on ∂Ω where p(x) ∈ C1 (ℝN) is a radial symmetric function such that sup |∇p(x)| < ∞, 1 < inf p(x) ≤ sup p(x) < ∞, and where -Δp(x)u = -div |∇u|p(x)-2 ∇u which is called the p(x)-Laplacian. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on ƒ(0), h(0).