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A generalised Nehari manifold method for a class of non linear Schrödinger systems in $\mathbb{R}^3$

Tommaso Cortopassi, Vladimir Georgiev

Abstract

We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schrödinger equations (NLSEs), that is $-ε^2 Δu + V(x) u= h_v(u,v), - ε^2 Δv + V(x) v=h_u (u,v)$. Under certain hypotheses on the potential $V$ and the non linearity $h$, we manage to prove that there exists a solution $(u_ε,v_ε)$ that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as $ε\to 0$. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is $\mathbb{R}^3$.

A generalised Nehari manifold method for a class of non linear Schrödinger systems in $\mathbb{R}^3$

Abstract

We study the existence of positive solutions of a particular elliptic system in composed of two coupled non linear stationary Schrödinger equations (NLSEs), that is . Under certain hypotheses on the potential and the non linearity , we manage to prove that there exists a solution that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as . We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is .
Paper Structure (5 sections, 13 theorems, 81 equations)

This paper contains 5 sections, 13 theorems, 81 equations.

Table of Contents

  1. INTRODUCTION
  2. TECHNICAL LEMMAS
  3. GENERALISED NEHARI MANIFOLD
  4. PROOF OF THE MAIN THEOREM
  5. ACKNOWLEDGEMENTS

Key Result

Theorem 1

Main Theorem Under hypotheses V1), V2), h1)-h6) there exists $\epsilon_0 >0$ such that for every $0 < \epsilon < \epsilon_0$ the system (main system) admits classical positive solutions $u_\epsilon, v_\epsilon \in C^2(\mathbb{R}^3) \cap H^2 (\mathbb{R}^3)$ with locally hölderian second derivatives, for every $i=1, \dots , k$ and for some positive constants $b,\gamma, \beta$. The uniqueness of loc

Theorems & Definitions (28)

  • Remark 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 2
  • Lemma 4
  • ...and 18 more