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

Part of: Elliptic equations and systems Partial differential equations

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)
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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
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

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