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), Talk no. 21, 10 p.

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

$$\left(a+b\sum _{j=1}^{p}{\int}_{M}{\left|\nabla {u}_{j}\right|}^{2}d{v}_{g}\right){\Delta}_{g}{u}_{i}+\sum _{j=1}^{p}{A}_{ij}{u}_{j}={\left|U\right|}^{{2}^{\u2606}-2}{u}_{i}$$ |

for all $i=1,\cdots ,p$, where ${\Delta}_{g}$ is the Laplace-Beltrami operator, $A$ is a ${C}^{1}$-map from $M$ into the space ${M}_{s}^{p}\left(\mathbb{R}\right)$ of symmetric $p\times p$ matrices with real entries, the ${A}_{ij}$’s are the components of $A$, $U=({u}_{1},\cdots ,{u}_{p})$, $\left|U\right|:M\to \mathbb{R}$ is the Euclidean norm of $U$, ${2}^{\u2606}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and we require that ${u}_{i}\ge 0$ in $M$ for all $i=1,\cdots ,p$. 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 $A$.

Author's affiliations:
Emmanuel Hebey ^{1};
Pierre-Damien Thizy ^{1}
^{1} Université de Cergy-Pontoise CNRS Département de Mathématiques F-95000 Cergy-Pontoise France

@article{SLSEDP_2013-2014____A21_0, author = {Emmanuel Hebey and Pierre-Damien Thizy}, title = {Recent results on stationary critical {Kirchhoff} systems in closed manifolds}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:21}, pages = {1--10}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2013-2014}, doi = {10.5802/slsedp.64}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.64/} }

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Emmanuel Hebey; Pierre-Damien Thizy. Recent results on stationary critical Kirchhoff systems in closed manifolds. Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 21, 10 p. doi : 10.5802/slsedp.64. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.64/

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