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Least action nodal solutions for the quadratic Choquard equation


Authors: Marco Ghimenti, Vitaly Moroz and Jean Van Schaftingen
Journal: Proc. Amer. Math. Soc. 145 (2017), 737-747
MSC (2010): Primary 35J91; Secondary 35J20, 35Q55
DOI: https://doi.org/10.1090/proc/13247
Published electronically: August 17, 2016
MathSciNet review: 3577874
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Abstract: We prove the existence of a minimal action nodal solution for the quadratic Choquard equation

$\displaystyle -\Delta u + u = \bigl (I_\alpha \ast \Vert u\Vert^2\bigr )u$$\displaystyle \quad \text {in \(\mathbb{R}^N\)},$    

where $ I_\alpha $ is the Riesz potential of order $ \alpha \in (0,N)$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations

$\displaystyle -\Delta u + u = \bigl (I_\alpha \ast \Vert u\Vert^p\bigr )\vert u\vert^{p-2}u$$\displaystyle \quad \text {in \(\mathbb{R}^N\)}$    

when $ p\searrow 2$. The existence of minimal action nodal solutions for $ p>2$ can be proved using a variational minimax procedure over a Nehari nodal set. No minimal action nodal solutions exist when $ p<2$.

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Additional Information

Marco Ghimenti
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy
Email: marco.ghimenti@dma.unipi.it

Vitaly Moroz
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales, United Kingdom
Email: V.Moroz@swansea.ac.uk

Jean Van Schaftingen
Affiliation: Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
Email: Jean.VanSchaftingen@UCLouvain.be

DOI: https://doi.org/10.1090/proc/13247
Keywords: Stationary nonlinear Schr\"odinger--Newton equation, stationary Hartree equation, nodal Nehari set, concentration-compactness
Received by editor(s): November 17, 2015
Received by editor(s) in revised form: April 17, 2016
Published electronically: August 17, 2016
Communicated by: Catherine Sulem
Article copyright: © Copyright 2016 American Mathematical Society