Nondegeneracy of Ground States and Multiple Semiclassical Solutions of the Hartree Equation for General Dimensions

Abstract

We study nondegeneracy of ground states of the Hartree equation

$$ -\Delta u+u=(I_{2}*u^2)u\quad \text{ in } {\mathbb {R}}^n $$

where \(n=3,4,5\) and \(I_2\) is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov–Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential

$$ -\varepsilon ^2\Delta u+u+V(x)u=\varepsilon ^{-2}(I_{2}*u^2)u\quad \text{ in } {\mathbb {R}}^n. $$

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Acknowledgements

The author is indebted to Minbo Yang for many useful discussions on Choquard equation. This work is supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY18A010023), NSFC (No.11771386) and First Class Discipline of Zhejiang - A (Zhejiang University of Finance and Economics- Statistics).

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Correspondence to Guoyuan Chen.

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Chen, G. Nondegeneracy of Ground States and Multiple Semiclassical Solutions of the Hartree Equation for General Dimensions. Results Math 76, 34 (2021). https://doi.org/10.1007/s00025-020-01332-y

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Keywords

  • Hartree equation
  • Schrödinger–Newton equation
  • nondegeneracy of ground states
  • semiclassical solutions
  • Lyapunov–Schmidt reduction

Mathematics Subject Classification

  • 35J61
  • 35J91
  • 35Q40