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 as
for 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 .
@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/} }
TY - JOUR AU - Emmanuel Hebey AU - Pierre-Damien Thizy TI - Recent results on stationary critical Kirchhoff systems in closed manifolds JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:21 PY - 2013-2014 SP - 1 EP - 10 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.64/ DO - 10.5802/slsedp.64 LA - en ID - SLSEDP_2013-2014____A21_0 ER -
%0 Journal Article %A Emmanuel Hebey %A Pierre-Damien Thizy %T Recent results on stationary critical Kirchhoff systems in closed manifolds %J Séminaire Laurent Schwartz — EDP et applications %Z talk:21 %D 2013-2014 %P 1-10 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.64/ %R 10.5802/slsedp.64 %G en %F SLSEDP_2013-2014____A21_0
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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