En théorie des points critiques, le théorème de Clark assure l'existence d'une suite de valeurs critiques négatives tendant vers 0 pour des fonctionnelles paires et coercitives. Nous étendons le théorème de Clark en montrant qu'une telle fonctionnelle possède une suite de points critiques tendant vers 0. Notre résultat permet aussi une description plus précise de l'ensemble des points critiques autour de l'origine. Une extension du théorème de Clark est aussi donnée. Nos résultats abstraits s'avèrent puissants dans les applications et conduisent à des résultats nouveaux concernant l'existence d'une infinité de solutions pour des problèmes partiellement sous linéaires comme des équations elliptiques ou des systèmes hamiltoniens.
In critical point theory, Clark's theorem asserts the existence of a sequence of negative critical values tending to 0 for even coercive functionals. We improve Clark's theorem, showing that such a functional has a sequence of critical points tending to 0. Our result also gives more detailed structure of the set of critical points near the origin. An extension of Clark's theorem is also given. Our abstract results are powerful in applications, and thus lead to much stronger results than those in the literature on existence of infinitely many solutions for partially sublinear problems such as elliptic equations and Hamiltonian systems.
Mots clés : Clark's theorem, Partially sublinear problem, Infinitely many solutions, Elliptic equation, Hamiltonian system
@article{AIHPC_2015__32_5_1015_0,
author = {Liu, Zhaoli and Wang, Zhi-Qiang},
title = {On {Clark's} theorem and its applications to partially sublinear problems},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1015--1037},
publisher = {Elsevier},
volume = {32},
number = {5},
year = {2015},
doi = {10.1016/j.anihpc.2014.05.002},
mrnumber = {3400440},
zbl = {1333.58004},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.002/}
}
TY - JOUR AU - Liu, Zhaoli AU - Wang, Zhi-Qiang TI - On Clark's theorem and its applications to partially sublinear problems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2015 SP - 1015 EP - 1037 VL - 32 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.002/ DO - 10.1016/j.anihpc.2014.05.002 LA - en ID - AIHPC_2015__32_5_1015_0 ER -
%0 Journal Article %A Liu, Zhaoli %A Wang, Zhi-Qiang %T On Clark's theorem and its applications to partially sublinear problems %J Annales de l'I.H.P. Analyse non linéaire %D 2015 %P 1015-1037 %V 32 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.002/ %R 10.1016/j.anihpc.2014.05.002 %G en %F AIHPC_2015__32_5_1015_0
Liu, Zhaoli; Wang, Zhi-Qiang. On Clark's theorem and its applications to partially sublinear problems. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 5, pp. 1015-1037. doi : 10.1016/j.anihpc.2014.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.002/
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