On Clark's theorem and its applications to partially sublinear problems
Liu, Zhaoli ; Wang, Zhi-Qiang
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1015-1037 / Harvested from Numdam
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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.

Publié le : 2015-01-01
DOI : https://doi.org/10.1016/j.anihpc.2014.05.002
Classification:  35A15,  58E05,  70H05 
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     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},
     volume = {32},
     year = {2015},
     pages = {1015-1037},
     doi = {10.1016/j.anihpc.2014.05.002},
     mrnumber = {3400440},
     zbl = {1333.58004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_5_1015_0}
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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) pp. 1015-1037. doi : 10.1016/j.anihpc.2014.05.002. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_5_1015_0/

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