Semi-classical states for the Choquard equation

By Vitaly Moroz, Jean Van Schaftingen. Calculus of Variations and Partial Differential Equations, 2013

227 citations recorded by the source corpus.

We study the nonlocal equation -ε2Δuε+Vuε=ε-α(Iα∗|uε|p)|uε|p-2uεinRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\varepsilon ^2 \Delta u_\varepsilon +V u_\varepsilon = \varepsilon ^{-\alpha }\bigl (I_\alpha *|u_\varepsilon |^p\bigr ) |u_\varepsilon |^{p - 2} u_\varepsilon \quad \text {in }{\mathbb {R}}^N, \end{aligned}$$\end{document}where N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 1$$\end{document}, α∈(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0, N)$$\end{document}, Iα(x)=Aα/|x|N-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\alpha (x) = A_\alpha /|x |^{N - \alpha }$$\end{document} is the Riesz potential and ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} is a small parameter. We show that if the external potential V∈C(RN;[0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \in C ({\mathbb {R}}^N; [0, \infty ))$$\end{document} has a local minimum and p∈[2,(N+α)/(N-2)+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in [2, (N + \alpha )/(N - 2)_+)$$\end{document} then for all small ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} the problem has a family of solutions concentrating to the local minimum of V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V$$\end{document} provided that: either p>1+max(α,α+22)/(N-2)+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1 + \max (\alpha , \frac{\alpha + 2}{2})/(N - 2)_+$$\end{document}, or p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p > 2$$\end{document} and lim inf|x|→∞V(x)|x|2>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\liminf _{|x | \rightarrow \infty } V (x) |x |^2 > 0$$\end{document}, or p=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = 2$$\end{document} and infx∈RNV(x)(1+|x|N-α)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf _{x \in {\mathbb {R}}^N} V (x) (1 + |x |^{N-\alpha })>0$$\end{document}. Our assumptions on the decay of V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V$$\end{document} and admissible range of p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 2$$\end{document} are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/86c531325f69035de834f3357d08e61c1517379c

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Citations in the corpus

Mass minimizers and concentration for nonlinear Choquard equations in $\R^N$ (2015, background): “The concentration phenomena have also been studied in [21] and [4] by considering semiclassical limit of the Choquard equation −ε(2)∆u+ V u = ε(Iα ∗ |u|)|u|u in R .”
Singularly perturbed critical Choquard equations (2016, background): “1), Moroz and Van Schaftingen in [26] developed a nonlocal penalization technique and showed that equation (1.”
Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth (2018, methodology): “In [44], Moroz and Van Schaftingen established the existence of ground state solutions to (1.4) in the optimal range N + α N p N + α N − 2 .”