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Journal of Dynamical and Control Systems

, Volume 25, Issue 1, pp 79–94 | Cite as

Existence and Multiplicity of Standing Wave Solutions for a Class of Quasilinear Schrödinger Systems in \(\mathbb {R}^{N}\)

  • Hongxue SongEmail author
  • Caisheng Chen
  • Wei Liu
Article
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Abstract

In this paper, we study the following quasilinear Schrödinger systems of the form
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{l} -\triangle_{p} u_{j}+a_{j}(x)|u_{j}|^{p-2}u_{j}\,-\,\triangle_{p} (|u_{j}|^{2})u_{j}\,=\,\mu_{j}|u_{j}|^{q-2}u_{j}\,+\,\frac{1}{2}{\sum}_{i\neq j}\beta_{ij}|u_{i}|^{m}|u_{j}|^{m-2}u_{j}, ~x\!\in\! \mathbb{R}^{N},\\ u_{j}(x)\rightarrow 0,~\text{as} ~|x|\rightarrow \infty, \ j = 1,\ \cdots,\ k, \end{array} \right. \end{array} $$
(0.1)
where N ≥ 3, 2 ≤ pN, and the potential aj(x) is positive and bounded in \(\mathbb {R}^{N}\), μj > 0, βij = βji for 1 ≤ i < jk(k ≥ 2). Using symmetric mountain pass lemma, we obtain infinitely many solutions to Schrödinger system (0.1).

Keywords

Quasilinear Schrödinger systems Dual approach Mountain pass lemma 

Mathematics Subject Classification (2010)

35J50 35J70 35J92 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.College of ScienceNanjing University of Posts and TelecommunicationsNanjingPeople’s Republic of China

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