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Science China Mathematics

, Volume 62, Issue 3, pp 509–534 | Cite as

Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems with continuous potentials

  • Lushun WangEmail author
Articles
  • 41 Downloads

Abstract

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:
$$\left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + \left( {1 + \delta P\left( x \right)} \right)u = {\mu _1}{u^3} + \beta u{v^2}in\Omega , \hfill \\ - {\varepsilon ^2}\Delta v + \left( {1 + \delta Q\left( x \right)} \right)u = {\mu _2}{u^3} + \beta {u^2}vin\Omega , \hfill \\ u > 0,v > 0in\Omega , \hfill \\ \frac{{\partial u}}{{\partial v}} = \frac{{\partial u}}{{\partial v}} = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$
where Ω is a smooth bounded domain in ℝN for N = 2, 3, δ, ε, μ1 and μ2 are positive parameters, β ∈ ℝ, P(x) and Q(x) are two smooth potentials defined on \(\overline{\Omega}\), the closure of Ω. Due to Liapunov-Schmidt reduction method, we prove that (Aε) has at least O(1/(ε|lnε|)N) synchronized and O(1/(ε|lnε|)2N) segregated vector solutions for ε and δ small enough and some β ∈ ℝ. Moreover, for each m ∈ (0,N) there exist synchronized and segregated vector solutions for (Aε) with energies in the order of εN-m. Our results extend the result of Lin et al. (2007) from the Lin-Ni-Takagi problem to the nonlinear Schrödinger elliptic systems with continuous potentials.

Keywords

nonlinear coupled elliptic system Liapunov-Schmidt reduction methods synchronized and segregated vector solutions 

MSC(2010)

35J60 35J20 

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina

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