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.

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.

DOI : 10.1016/j.anihpc.2014.05.002
Classification : 35A15, 58E05, 70H05
Mots-clés : Clark's theorem, Partially sublinear problem, Infinitely many solutions, Elliptic equation, Hamiltonian system
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     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},
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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) 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/

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

[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

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

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