Multiple Solutions for a Singular Quasilinear Elliptic System

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system -div(|x|(-ap)|∇u|(p-2)∇u) + f₁(x)|u|(p-2) u = (α/(α + β))g(x)|u| (α-2) u|v| (β) + λh₁(x)|u| (γ-2) u + l₁(x), -div(|x|(-ap) |∇v| (p-2)∇v) + f₂(x)|v| (p-2) v = (β/(α + β))g(x)|v|(β-2) v|u|(α) + μh 2(x)|v|(γ-2)v + l 2(x), u(x) > 0, v(x) > 0, x ∈ ℝ(N), where λ, μ > 0, 1 < p < N, 1 < γ < p < α + β < p* = Np/(N-pd), 0 ≤ a < (N - p)/p, a ≤ b < a + 1, d = a + 1 - b > 0. The functions f₁(x), f₂(x), g(x), h₁(x), h₂(x), l₁(x), and l₂(x) satisfy some suitable conditions. We will prove that the problem has at least two nontrivial solutions by using Mountain Pass Theorem and Ekeland's variational principle.