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On Stable Weak Solutions of the Weighted Static Choquard Equation Involving Grushin Operator

On Stable Weak Solutions of the Weighted Static Choquard Equation Involving Grushin Operator In this paper we study solutions, possibly unbounded and sign-changing, of a weighted static Choquard equation involving the Grushin operator. Under some appropriate assumptions on the parameters, we prove various Liouville-type theorems for weak solutions under the assumption that they are stable or stable outside a compact set of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathbb{R}^{n}$\end{document}. First, we establish the standard integral estimates via stability property to derive the nonexistence results for stable weak solutions. Next, by means of the Pohozaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

On Stable Weak Solutions of the Weighted Static Choquard Equation Involving Grushin Operator

Rahal, Belgacem; Le, Phuong
Acta Applicandae Mathematicae , Volume 185 (1) – Jun 1, 2023
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18 pages

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Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-023-00570-w
Publisher site
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Abstract

In this paper we study solutions, possibly unbounded and sign-changing, of a weighted static Choquard equation involving the Grushin operator. Under some appropriate assumptions on the parameters, we prove various Liouville-type theorems for weak solutions under the assumption that they are stable or stable outside a compact set of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathbb{R}^{n}$\end{document}. First, we establish the standard integral estimates via stability property to derive the nonexistence results for stable weak solutions. Next, by means of the Pohozaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Jun 1, 2023

Keywords: Choquard equation; Weighted equation; Liouville-type theorems; Grushin operator; Stable solutions; Stability outside a compact set; Pohozaev identity

References

  • Liouville theorems for semilinear equations on the Heisenberg group
    Birindelli, I.; Capuzzo Dolcetta, I.; Cutrì, A.
  • On stable entire solutions of semi-linear elliptic equations with weights
    Cowan, C.; Fazly, M.
  • Finite Morse index solutions of an elliptic equation with supercritical exponent
    Dancer, E.N.; Du, Y.H.; Guo, Z.M.
  • Liouville-type theorem for a subelliptic equation with Choquard nonlinearity and weight
    Duong, A.T.; Nguyen, T.Q.
  • On the classification of solutions of the Lane–Emden equation on unbounded domains of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document}
    Farina, A.
  • Weighted Sobolev-Poincaré inequalities for Grushin type operators
    Franchi, B.; Gutiérrez, C.E.; Wheeden, R.L.
  • Stable solutions to the static Choquard equation
    Le, P.
  • Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator
    Le, P.
  • On classical solutions to the Hartree equation
    Le, P.
  • Liouville theorems for a p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-Laplace equation with Hartree type nonlinearity
    Le, P.
  • Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators
    Monticelli, D.D.
  • Liouville-type theorem for finite Morse index solutions to the Choquard equation involving Δλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta_{\lambda}$\end{document}-Laplacian
    Quyet, D.T.
  • Some Liouville theorems for Hénon type equations in half-space with nonlinear boundary value conditions and finite Morse indices
    Rahal, B.
  • Liouville-type theorems with finite Morse index for semilinear Δλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta _{\lambda}$\end{document}-Laplace operators
    Rahal, B.
  • On stable entire solutions of a sub-elliptic system involving advection terms with negative exponents and weights
    Rahal, B.
  • Some Liouville theorems for Hénon type elliptic equations
    Wang, C.; Ye, D.
  • Liouville-type theorem for nonlinear elliptic equations involving p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-Laplace-type Grushin operators
    Wei, Y.; Chen, C.; Chen, Q.; Yang, H.
  • On stable solutions of the weighted Lane-Emden equation involving Grushin operator
    Wei, Y.; Yang, H.; Yu, H.
  • Liouville type theorem for nonlinear elliptic equation involving Grushin operators
    Yu, X.
  • Liouville theorem for Choquard equation with finite Morse indices
    Zhao, X.
  • Liouville type theorem for some Choquard type equation
    Zhao, X.; Yu, X.