Hostname: page-component-5d84bcc8dc-z9l2z Total loading time: 0 Render date: 2026-09-08T08:15:05.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Groundstates for Choquard type equations with Hardy–Littlewood–Sobolev lower critical exponent

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  29 January 2019

Daniele Cassani ,
Jean Van Schaftingen  and
Jianjun Zhang
Daniele Cassani
Affiliation:
Dip. di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria and RISM - Riemann International School of Mathematics, via G.B. Vico 46, Varese21100, Italy (Daniele.Cassani@uninsubria.it)
Jean Van Schaftingen
Affiliation:
Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.01, Louvain-la-Neuve 1348, Belgium (Jean.VanSchaftingen@UCLouvain.be)
Jianjun Zhang
Affiliation:
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing400074, PR China and Dip. di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, via G.B. Vico 46, Varese21100, Italy (zhangjianjun09@tsinghua.org.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,

$$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$
where $N\ges 3$, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and $I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided $N \ges 4$ or in dimension N = 3 if in addition 3/2 < α < 3 and $\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.

Keywords

Ground statesnonlocal PDEexistence and nonexistenceChoquard equationHardy–Littlewood–Sobolev inequalitycritical growth

MSC classification

Primary: 35B33: Critical exponents
Secondary: 35J61: Semilinear elliptic equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1377 - 1400
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Bartolo, P., Benci, V. and Fortunato, D.. Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity. Nonlinear. Anal. Theory. Methods. Appl. 7 (1983), 981–1012.CrossRefGoogle Scholar
2Cassani, D. and Zhang, J. J.. Groundstates and semiclassical states of Choquard type equations involving Hardy–Littlewood–Sobolev critical growth. Adv. Nonlinear. Anal. (in press), doi:10.1515/anona-2018-0019.Google Scholar
3Donsker, M. D. and Varadhan, S. R. S.. Asymptotics for the polaron. Comm. Pure Appl. Math. XXXVI (1983), 505–528.CrossRefGoogle Scholar
4Elgart, A. and Schlein, B.. Mean field dynamics of Boson stars. Comm. Pure Appl. Math. 60 (2007), 500–545.CrossRefGoogle Scholar
5Lieb, E. H.. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57 (1976/77), 93–105.CrossRefGoogle Scholar
6Lieb, E. H. and Loss, M.. Analysis. Graduate studies in Mathematics, 2nd edn, vol. 14 (Providence: American Mathematical Society, 2001).Google Scholar
7Lions, P. L.. The Choquard equation and related questions. Nonlinear Anal. TMA 4 (1980), 1063–1073.CrossRefGoogle Scholar
8Lions, P. L.. Compactness and topological methods for some nonlinear variational problems of mathematical physics, Nonlinear problems: present and future 17–34, (1982).CrossRefGoogle Scholar
9Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case I. II. Annales Inst. H. Poincaré Analyse Non Linéaire 1 (1984), 109–145, 223–283.CrossRefGoogle Scholar
10Mercuri, C., Moroz, V. and Van Schaftingen, J.. Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency. Calc. Var. Partial Differ. Equ. 55 (2016), no. 146, 1–58.CrossRefGoogle Scholar
11Moroz, V. and Van Schaftingen, J.. Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265 (2013), 153–184.CrossRefGoogle Scholar
12Moroz, V. and Van Schaftingen, J.. Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent. Comm. Contem. Math. 17 (2015), 1550005 (12 pages).CrossRefGoogle Scholar
13Moroz, V. and Van Schaftingen, J.. A guide to the Choquard equation. J. Fixed Point Theory Appl. 19 (2017), 773–813.CrossRefGoogle Scholar
14Pekar, S.. Untersuchung über die Elektronentheorie der Kristalle (Berlin: Akademie Verlag, 1954).Google Scholar
15Penrose, R.. On gravity’s role in quantum state reduction. Gen. Rel. Grav. 28 (1996), 581–600.CrossRefGoogle Scholar
16Simon, B.. Schrödinger semigroups. Bull. Amer. Math. Soc. 7 (1982), 447–526.CrossRefGoogle Scholar