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

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 R N for N = 2, 3, δ, e, μ1 and μ2 are positive parameters, β ∈ R, P(x) and Q(x) are two smooth potentials defined on Ω, the closure of Ω. Due to Liapunov-Schmidt reduction method, we prove that (Ae) has at least O(1/(e|lne|) N ) synchronized and O(1/(e|lne|)2N) segregated vector solutions for e and δ small enough and some β ∈ R. Moreover, for each m ∈ (0,N) there exist synchronized and segregated vector solutions for (Ae) with energies in the order of eN-m. Our results extend the result of Lin et al. (2007) from the Lin-Ni-Takagi problem to the nonlinear Schr¨odinger elliptic systems with continuous potentials.