Multiple Solutions for a Singular Quasilinear Elliptic System

摘要

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system −div(|x|^−ap|∇u|^p−2∇u) + f 1(x)|u|^p−2 u = (α/(α + β))g(x)|u|^α−2 u|v|^β + λh 1(x)|u|^γ−2 u + l 1(x), −div(|x|^−ap|∇v|^p−2∇v) + f 2(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 f 1(x), f 2(x), g(x), h 1(x), h 2(x), l1(x), andl2(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.