Normalized Solutions of the Choquard Equation with Sobolev Critical Exponent

Abstract

In this paper, we consider the existence and asymptotic behavior normalized solutions for the following Choquard equation involving Sobolev critical exponent

-\mathrm{\Delta }u=\lambda u+(I_{\alpha }\ast \mid u{\mid }^q)\mid u{\mid }^{q-2}u+\mid u{\mid }^{4}u\mathrm{i}\mathrm{n}{\mathbb{R}}^3,

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\Delta{u}=\lambda{u}+(I_{\alpha} \ast \ |u|^{q})|u|^{q-2}{u}+|u|^{4}{u}\quad{\rm in} \ {\mathbb R}^{3},$$\end{document}

under the prescribed L²-norm

{\int }_{{\mathbb{R}}^3}{u}^2=c^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\int_{\mathbb R^{3}} \ u^{2}=c^{2}$$\end{document}

with c > 0, where I_α denotes the Riesz potential. Let

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${5 \over 3} < q < 3$$\end{document}

. When α > 0 small enough, we obtain the existence of the positive ground state solutions, which converge to a least energy solution of the limiting critical local problem as α → 0⁺.

Keywords

Choquard problem / critical exponent / normalized solutions / asymptotic behavior

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