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

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## Abstract

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials: where Ω is a smooth bounded domain in ℝ

$$\left\{ \begin{gathered}
- {\varepsilon ^2}\Delta u + \left( {1 + \delta P\left( x \right)} \right)u = {\mu _1}{u^3} + \beta u{v^2}in\Omega , \hfill \\
- {\varepsilon ^2}\Delta v + \left( {1 + \delta Q\left( x \right)} \right)u = {\mu _2}{u^3} + \beta {u^2}vin\Omega , \hfill \\
u > 0,v > 0in\Omega , \hfill \\
\frac{{\partial u}}{{\partial v}} = \frac{{\partial u}}{{\partial v}} = 0on\partial \Omega , \hfill \\
\end{gathered} \right.$$

^{N}for*N*= 2, 3,*δ*,*ε*,*μ*_{1}and*μ*_{2}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 (*A*_{ε}) has at least*O*(1/(*ε*|ln*ε*|)^{N}) synchronized and*O*(1/(*ε*|ln*ε*|)^{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 (*A*_{ε}) with energies in the order of*ε*^{N-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## Preview

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