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Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency

  • Carlo Mercuri
  • Vitaly Moroz
  • Jean Van Schaftingen
Article

Abstract

We study the nonlocal Schrödinger–Poisson–Slater type equation
$$\begin{aligned} - \Delta u + (I_\alpha *\vert u\vert ^p)\vert u\vert ^{p - 2} u= \vert u\vert ^{q-2}u\quad \text {in}\quad \mathbb {R}^N, \end{aligned}$$
where \(N\in \mathbb {N}\), \(p>1\), \(q>1\) and \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0,N).\) We introduce and study the Coulomb–Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

Mathematics Subject Classification

35Q55 (35J91, 35J47, 35J50, 31B35) 

Notes

Acknowledgments

C.M. would like to thank Antonio Ambrosetti for drawing his attention to several questions related to nonlinear Schrödinger-Poisson systems. J.V.S was funded by the Fonds de la Recherche Scientifique—FNRS (Grant T.1110.14)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Carlo Mercuri
    • 1
  • Vitaly Moroz
    • 1
  • Jean Van Schaftingen
    • 2
  1. 1.Department of MathematicsSwansea UniversitySwanseaWales, UK
  2. 2.Institut de Recherche en Mathématique et PhysiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium

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