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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) + 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 (...

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Detalles Bibliográficos
Autores principales: Chen, Lin, Chen, Caisheng, Xiu, Zonghu
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Hindawi Publishing Corporation 2013
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3824403/
https://www.ncbi.nlm.nih.gov/pubmed/24282377
http://dx.doi.org/10.1155/2013/278013
Descripción
Sumario: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), l (1)(x), and l (2)(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.