On Clark's theorem and its applications to partially sublinear problems
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, p. 1015-1037
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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.

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.

DOI : https://doi.org/10.1016/j.anihpc.2014.05.002
Classification:  35A15,  58E05,  70H05
Keywords: 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},
publisher = {Elsevier},
volume = {32},
number = {5},
year = {2015},
pages = {1015-1037},
doi = {10.1016/j.anihpc.2014.05.002},
zbl = {1333.58004},
mrnumber = {3400440},
language = {en},
url = {http://www.numdam.org/item/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, Volume 32 (2015) no. 5, pp. 1015-1037. doi : 10.1016/j.anihpc.2014.05.002. http://www.numdam.org/item/AIHPC_2015__32_5_1015_0/

[1] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519 -543 | MR 1276168 | Zbl 0805.35028

[2] T. Bartsch, Z. Liu, Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. Differ. Equ. 9 (2004), 645 -676 | MR 2099975 | Zbl 1100.58005

[3] T. Bartsch, M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Am. Math. Soc. 123 (1995), 3555 -3561 | MR 1301008 | Zbl 0848.35039

[4] V. Benci, A new approach to the Morse–Conley theory and some applications, Ann. Mat. Pura Appl. 158 (1991), 231 -305 | MR 1131853 | Zbl 0778.58011

[5] K.-C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102 -129 | MR 614246 | Zbl 0487.49027

[6] D.C. Clark, A variant of the Lusternik–Schnirelman theory, Indiana Univ. Math. J. 22 (1972–1973), 65 -74 | MR 296777

[7] J.-N. Corvellec, On the second deformation lemma, Topol. Methods Nonlinear Anal. 17 (2001), 55 -66 | MR 1846978 | Zbl 0990.58010

[8] M. Degiovanni, On topological and metric critical point theory, J. Fixed Point Theory Appl. 7 (2010), 85 -102 | MR 2652512 | Zbl 1205.58007

[9] M. Degiovanni, M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl. 167 (1994), 73 -100 | MR 1313551 | Zbl 0828.58006

[10] J. Garcia Azorero, I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Am. Math. Soc. 323 (1991), 877 -895 | MR 1083144 | Zbl 0729.35051

[11] H.P. Heinz, Free Ljusternik–Schnirelman theory and the bifurcation diagrams of certain singular nonlinear systems, J. Differ. Equ. 66 (1987), 263 -300 | MR 871998 | Zbl 0607.34012

[12] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352 -370 | MR 2152503 | Zbl 1081.49002

[13] J. Liu, Y. Guo, Critical point theory for nonsmooth functionals, Nonlinear Anal. 66 (2007), 2731 -2741 | MR 2311634 | Zbl 1117.58008

[14] Z. Liu, Z.-Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys. 56 (2005), 609 -629 | MR 2185298 | Zbl 1113.35071

[15] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. vol. 74 , Springer-Verlag, New York (1989) | MR 982267 | Zbl 0676.58017

[16] R.S. Palais, Critical point theory and the minimax principle, Global Analysis, Proc. Symp. Pure Math. vol. 15 , AMS, Providence, RI (1970), 185 -212 | MR 264712 | Zbl 0212.28902

[17] M. Struwe, Variational Methods, Springer-Verlag, Berlin (1996) | MR 1411681

[18] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. vol. 65 , AMS, Providence (1986) | MR 845785

[19] Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differ. Equ. Appl. 8 (2001), 15 -33 | MR 1828946 | Zbl 0983.35052