On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u _i|^p-2∇u_i)+λk_i(|x|) fⁱ(u₁, . . . , u_n) = 0, p > 1, R₁ < |x| < R₂, u_i(x) = 0, on |x| = R₁ and R₂, i = 1, . . . , n, x ∈ ℝ^N, where k _i and fⁱ, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u₁, . . . , u_n), φ(t) = |t| ^p-2t, f₀ⁱ = lim_∥u∥→0o fⁱ(u)/φ(∥u∥), f_∞ⁱ = lim _{∥u∥→∞} fⁱ(u)/φ(∥u∥), i = 1, . . . , n, f = (f¹ , . . . , fⁿ), f₀ = Σ_i=1ⁿ f₀ⁱ and f _∞ = Σ_i=1ⁿ f_∞ ⁱ. We prove that either f₀ = 0 and f_∞ = ∞ (superlinear), or f₀ = ∞ and f_∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f ⁱ(u) > 0 for ∥u∥ > 0, i = 1, . . . , n, then either f₀ = f_∞, = 0, or f₀ = f_∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all die results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point meorems in a cone.