Multiple positive solutions for a quasilinear system of Schrödinger equations

Article

Abstract.

We consider the quasilinear system
$$ \left\{ \begin{gathered} - \varepsilon ^{p} \,div\,(|\nabla u|^{{p - 2\,\,}} \nabla u)\, + \,V\,(z)\,u\,^{{p - 1\,}} \, = \,\,Q_{{u\,\,}} (u,v)\, + \,\gamma H_{u} \,(u,v)\,in\,\mathbb{R}^{N} \,, \hfill \\ - \varepsilon ^{p} \,div\,(|\nabla v|^{{p - 2\,\,}} \nabla v)\, + \,W\,(z)\,v\,^{{p - 1\,}} \, = \,\,Q_{{v\,\,}} (u,v)\, + \,\gamma H_{v} \,(u,v)\,{\rm in}\,\mathbb{R}^{N} \,, \hfill \\ u,\,v\, \in \,W\,^{{1,p\,}} (\mathbb{R}^{N} ),\,\,\,\,\,u(z)\,,\,v(z)\, > \,0\,for\,all\,z {\rm in} \mathbb{R}^{{N\,}} ,\, \hfill \\ \end{gathered} \right.$$
where \(\varepsilon > 0, 2 \leq p < N\), V and W are positive continuous potentials, Q is an homogeneous function with subcritical growth, \(H(u, v) = |u|^{\alpha}|v|^{\beta}\) with \(\alpha,\beta \geq 1\) satisfying \(\alpha + \beta = Np/(N - p)\). We relate the number of solutions with the topology of the set where V and W attain it minimum values. We consider the subcritical case γ = 0 and the critical case γ = 1. In the proofs we apply Ljusternik-Schnirelmann theory.

Keywords:

Quasilinear Schrödinger equation Ljusternik-Schnirelmann theory positive solutions critical problems 

Mathematics Subject Classification (2000).

Primary 35J50 Secondary 35B33, 58E05 

Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do ParáBelémBrazil
  2. 2.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

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