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Mediterranean Journal of Mathematics

, Volume 11, Issue 3, pp 891–903 | Cite as

On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

  • G. A. Afrouzi
  • Nguyen Thanh Chung
  • Z. Naghizadeh
Article

Abstract

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system
$$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$
where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C 1-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.

Mathematics Subject Classification (2000)

35B30 35J60 35P15 

Keywords

Quasilinear elliptic systems Multiplicity Nonexistence Variational methods 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • G. A. Afrouzi
    • 1
  • Nguyen Thanh Chung
    • 2
  • Z. Naghizadeh
    • 1
  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran
  2. 2.Department of Mathematics and InformaticsQuang Binh UniversityDong Hoi, Quang BinhVietnam

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