An existence theorem for quasilinear systems

This paper deals with the existence of positive radial solutions for the quasilinear system div(|∇_ui|^p-2∇_ui + λfⁱ (u₁,...,u_n) = 0, |x| < 1, u_i(x) = 0, on |x| = 1, i = 1,...,n p > 1, λ > 0, x ∈ ℝ^N. The fⁱ, i = 1,..., n, are continuous and non-negative functions. Let u = (u₁,...,u_n), ||u|| = Σ_i=1ⁿ |u_i|, f₀ⁱ= lim/||u||→0 fⁱ(u)/||u||^p-1 i = 1,..., n, f = (f ¹,..., fⁿ), f₀ = Σ_i=1 ⁿ f₀ⁱ. We prove that the problem has a positive solution for sufficiently small λ > 0 if f₀ = ∞. Our methods employ a fixed-point theorem in a cone.