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Partially concentrating standing waves for weakly coupled Schrödinger systems

Benedetta Pellacci, Angela Pistoia, Giusi Vaira, Gianmaria Verzini

Abstract

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}ψ_{1}=-\frac{\hslash^2}{2m_{1}}Δψ_{1}+ {V_1}(x)ψ_{1}-μ_{1}|ψ_{1}|^{2}ψ_{1}-β|ψ_{2}|^{2}ψ_{1} & \\ i \hslash \partial_{t}ψ_{2}=-\frac{\hslash^2}{2m_{2}}Δψ_{2}+ {V_2}(x)ψ_{2}-μ_{2}|ψ_{2}|^{2}ψ_{2}-β|ψ_{1}|^{2}ψ_{2},& \end{cases} \] where $V_1$ and $V_2$ are radial potentials bounded from below. We address the case $m_{1}\sim \hslash^2\to0$, $m_2$ constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.

Partially concentrating standing waves for weakly coupled Schrödinger systems

Abstract

We study the existence of standing waves for the following weakly coupled system of two Schrödinger equations in , , where and are radial potentials bounded from below. We address the case , constant, and prove the existence of a standing wave solution with both nontrivial components satisfying a prescribed asymptotic profile. In particular, the second component of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature.
Paper Structure (9 sections, 10 theorems, 141 equations)

This paper contains 9 sections, 10 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. The ansatz and the correction terms.
  4. Rewriting the problem
  5. The linear theory
  6. The size of the error term
  7. Solving the second equation in \ref{['eq:sistpro']}
  8. Solving the reduced problem
  9. Technical Lemma

Key Result

Theorem 1.1

Let $N=2,3$, and suppose that ${\bf (V_{1}), (V_{2})}$ and ${\bf (W)}$ hold. Assume $\beta<0$. Let set $\omega_0:=\omega(0)>0$ and assume that Then there exists $\varepsilon_{0}>0$ such that for every $\varepsilon\in (0,\varepsilon_{0})$ there exists a solution $(u_{\varepsilon},v_{\varepsilon})$ of system pro:P0 even with respect to each variable and having the following asymptotic profile as $

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 21 more