Recent results on stationary critical Kirchhoff systems in closed manifolds

Emmanuel Hebey[1]; Pierre-Damien Thizy[1]

  • [1] Université de Cergy-Pontoise CNRS Département de Mathématiques F-95000 Cergy-Pontoise France

Séminaire Laurent Schwartz — EDP et applications (2013-2014)

  • Volume: 18, Issue: 2, page 1-10
  • ISSN: 2266-0607

Abstract

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We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let be a closed -manifold, . The critical Kirchhoff systems we consider are written asfor all , where is the Laplace-Beltrami operator, is a -map from into the space of symmetric matrices with real entries, the ’s are the components of , , is the Euclidean norm of , is the critical Sobolev exponent, and we require that in for all . We discuss the two following issues in this text: the question of the existence of nontrivial solutions to our systems, together with the dual question of getting nonexistence results in parallel to our existence results, and the question of the stability of our systems which measures how much the equations are robust with respect to variations of their natural parameters , , and .

How to cite

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Hebey, Emmanuel, and Thizy, Pierre-Damien. "Recent results on stationary critical Kirchhoff systems in closed manifolds." Séminaire Laurent Schwartz — EDP et applications 18.2 (2013-2014): 1-10. <http://eudml.org/doc/275703>.

@article{Hebey2013-2014,
abstract = {We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n \ge 3$. The critical Kirchhoff systems we consider are written as\begin\{equation*\} \Bigl (a + b\sum \_\{j=1\}^p\int \_M\vert \nabla u\_j\vert ^2dv\_g\Bigr )\Delta \_gu\_i + \sum \_\{j=1\}^pA\_\{ij\}u\_j = \left|U\right|^\{2^\star -2\}u\_i \end\{equation*\}for all $i = 1,\dots ,p$, where $\Delta _g$ is the Laplace-Beltrami operator, $A$ is a $C^1$-map from $M$ into the space $M^p_s(\mathbb\{R\})$ of symmetric $p\times p$ matrices with real entries, the $A_\{ij\}$’s are the components of $A$, $U = (u_1,\dots ,u_p)$, $\vert U\vert : M \rightarrow \mathbb\{R\}$ is the Euclidean norm of $U$, $2^\star = \frac\{2n\}\{n-2\}$ is the critical Sobolev exponent, and we require that $u_i \ge 0$ in $M$ for all $i = 1,\dots ,p$. We discuss the two following issues in this text: the question of the existence of nontrivial solutions to our systems, together with the dual question of getting nonexistence results in parallel to our existence results, and the question of the stability of our systems which measures how much the equations are robust with respect to variations of their natural parameters $a$, $b$, and $A$.},
affiliation = {Université de Cergy-Pontoise CNRS Département de Mathématiques F-95000 Cergy-Pontoise France; Université de Cergy-Pontoise CNRS Département de Mathématiques F-95000 Cergy-Pontoise France},
author = {Hebey, Emmanuel, Thizy, Pierre-Damien},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {blow-up theory; critical exponent; elliptic stability; Kirchhoff systems},
language = {eng},
number = {2},
pages = {1-10},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Recent results on stationary critical Kirchhoff systems in closed manifolds},
url = {http://eudml.org/doc/275703},
volume = {18},
year = {2013-2014},
}

TY - JOUR
AU - Hebey, Emmanuel
AU - Thizy, Pierre-Damien
TI - Recent results on stationary critical Kirchhoff systems in closed manifolds
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2013-2014
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 18
IS - 2
SP - 1
EP - 10
AB - We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n \ge 3$. The critical Kirchhoff systems we consider are written as\begin{equation*} \Bigl (a + b\sum _{j=1}^p\int _M\vert \nabla u_j\vert ^2dv_g\Bigr )\Delta _gu_i + \sum _{j=1}^pA_{ij}u_j = \left|U\right|^{2^\star -2}u_i \end{equation*}for all $i = 1,\dots ,p$, where $\Delta _g$ is the Laplace-Beltrami operator, $A$ is a $C^1$-map from $M$ into the space $M^p_s(\mathbb{R})$ of symmetric $p\times p$ matrices with real entries, the $A_{ij}$’s are the components of $A$, $U = (u_1,\dots ,u_p)$, $\vert U\vert : M \rightarrow \mathbb{R}$ is the Euclidean norm of $U$, $2^\star = \frac{2n}{n-2}$ is the critical Sobolev exponent, and we require that $u_i \ge 0$ in $M$ for all $i = 1,\dots ,p$. We discuss the two following issues in this text: the question of the existence of nontrivial solutions to our systems, together with the dual question of getting nonexistence results in parallel to our existence results, and the question of the stability of our systems which measures how much the equations are robust with respect to variations of their natural parameters $a$, $b$, and $A$.
LA - eng
KW - blow-up theory; critical exponent; elliptic stability; Kirchhoff systems
UR - http://eudml.org/doc/275703
ER -

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