A mass supercritical problem revisited

By Louis Jeanjean, Sheng-Sen Lu. Calculus of Variations and Partial Differential Equations, 2020

216 citations recorded by the source corpus.

In any dimension 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} and for given mass m>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>0$$\end{document}, we revisit the nonlinear scalar field equation with an L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} constraint: -Δu=f(u)-μuinRN,‖u‖L2(RN)2=m,u∈H1(RN),(Pm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=f(u)-\mu u\quad \text {in}~\mathbb {R}^N,\\ \Vert u\Vert ^2_{L^2(\mathbb {R}^N)}&=m,\\ u&\in H^1(\mathbb {R}^N), \end{aligned} \right. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (P_m) \end{aligned}$$\end{document}where μ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \mathbb {R}$$\end{document} will arise as a Lagrange multiplier. Assuming only that the nonlinearity f is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states to (Pm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_m)$$\end{document} and reveal the basic behavior of the ground state energy Em\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_m$$\end{document} as m>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>0$$\end{document} varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document} and establish the existence and multiplicity of nonradial sign-changing solutions when N≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 4$$\end{document}. Finally we propose two open problems.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3aaf40fdf12af0874cb76630133bb064934edf7f

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In Applied Mathematics

  1. Yuanze Wu
  2. Jun-cheng Wei
  3. Manuel del Pino
  4. A. Malchiodi
  5. Xinfu Li
  6. Shusen Yan
  7. Louis Jeanjean
  8. Shuangjie Peng

In Mathematical Physics

  1. Yuanze Wu
  2. Jun-cheng Wei
  3. Xinfu Li
  4. Manuel del Pino
  5. Zhi-Qiang Wang
  6. Louis Jeanjean
  7. Shuangjie Peng
  8. Thanh Trung Le

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