A critical point theorem for a class of non-differentiable functionals with applications

Yan Ning; Daowei Lu; Yan Ning; Daowei Lu

doi:10.3934/math.2020287

AIMS Mathematics

2020, Volume 5, Issue 5: 4466-4481. doi: 10.3934/math.2020287

Previous Article Next Article

Research article

A critical point theorem for a class of non-differentiable functionals with applications

Yan Ning ^,,
Daowei Lu

Department of Mathematics, Jining University, Qufu 273155, Shandong, P. R. China

Received: 10 January 2020 Accepted: 12 May 2020 Published: 19 May 2020
MSC : 35B38, 49J52

This paper presents a multiplicity theorem for a kind of non-smooth functionals. The proof of this theorem relies on a suitable deformation lemma and the perturbation methods. We also apply this result to prove a multiplicity theorem for elliptic variational-hemivariational inequality problems.
- critical points,
- non-smooth functions,
- deformation,
- variational-hemivariational inequality,
- locally Lipschitz continuous
Citation: Yan Ning, Daowei Lu. A critical point theorem for a class of non-differentiable functionals with applications[J]. AIMS Mathematics, 2020, 5(5): 4466-4481. doi: 10.3934/math.2020287

Related Papers:

Abstract

This paper presents a multiplicity theorem for a kind of non-smooth functionals. The proof of this theorem relies on a suitable deformation lemma and the perturbation methods. We also apply this result to prove a multiplicity theorem for elliptic variational-hemivariational inequality problems.

References

[1]	G. M. Bisci, Some remarks on a recent critical point result of nonsmooth analysis, Le Matematiche, 64 (2009), 97-112.
[2]	H. Brézis, Analyse Fonctionelle, Théorie et Applications, 1983.
[3]	H. Brézis, L. Nirenberg, Remarks on finding critical points, Commun. Pur. Appl. Math., 44 (1991), 939-961. doi: 10.1002/cpa.3160440808
[4]	P. Candito, R. Livrea, D. Motreanu, Bounded Palais-Smale sequences for non-differentiable functions, Nonlinear Anal-Theor., 74 (2011), 5446-5454. doi: 10.1016/j.na.2011.05.030
[5]	K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0
[6]	F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
[7]	N. Costea, C. Varga, Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions, J. Global Optim., 56 (2013), 399-416. doi: 10.1007/s10898-011-9801-3
[8]	K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[9]	J. Dugundji, Topology, Allyn and Bacon, Boston, 1996.
[10]	F. Faraci, A. Iannizzotto, Three nonzero periodic solutions for a differential inclusion, Discrete Cont. Dyn. S, 5 (2012), 779-788.
[11]	A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications, Nonlinear Anal-Theor., 72 (2010), 1319-1338. doi: 10.1016/j.na.2009.08.001
[12]	S. T. Kyritsi, N. S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities, Nonlinear Anal-Theor., 61 (2005), 373-403. doi: 10.1016/j.na.2004.12.001
[13]	S. T. Kyritsi, N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance, J. Math. Anal. Appl., 276 (2002), 292-313. doi: 10.1016/S0022-247X(02)00443-2
[14]	S. J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6-32. doi: 10.1006/jmaa.1995.1002
[15]	Z. Li, Y. Shen, Y. Zhang, An application of nonsmooth critical point theory, Topol. Method. Nonl. An., 35 (2010), 203-219.
[16]	R. Livrea, S. A. Marano, D. Motreanu, Critical points for nondifferential function in presence of splitting, J. Differ. Equations, 226 (2006), 704-725. doi: 10.1016/j.jde.2005.11.001
[17]	A. M. Mao, S. X. Luan, Periodic solutions of an infinite-dimensional Hamiltonian system, Appl. Math. Comput., 201 (2008), 800-804.
[18]	A. M. Mao, M. Xue, Positive solutions of singular boundary value problems, Acta Math. Sin., 44 (2001), 899-908.
[19]	A. M. Mao, Y. Chen, Existence and Concentration of Solutions For Sublinear Schrödinger-Poisson Equations, Indian J. Pure Ap. Mat., 49 (2018), 339-348. doi: 10.1007/s13226-018-0272-9
[20]	S. A. Marano, D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian, J. Differ. Equations, 182 (2002), 108-120. doi: 10.1006/jdeq.2001.4092
[21]	S. A. Marano, D. Motreanu, A deformation theorem and some critical point results for nondifferentiale functions, Topol. Method. Nonl. An., 22 (2003), 139-158.
[22]	D. Motreanu, P. D. Panagiotopoulos, Minimax Theorems and Qualitative properties of the solutions of the Hemivariational Inequalities, 1999.
[23]	D. Motreanu, V. Radulescu, Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Springer Science & Business Media, 2003.
[24]	P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer, Berlin, 1993.
[25]	A. Szulkin, Minimax principle for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. I. H. Poincare-An., 3 (1986), 77-109. doi: 10.1016/S0294-1449(16)30389-4
[26]	J. Wang, T. An, F. Zhang, Positive solutions for a class of quasilinear problems with critical growth in $\mathbb{R}^N$, P. Roy. Soc. Edinb. A, 145 (2015), 411-444.
[27]	Y. Wu, T. An, Existence of periodic solutions for non-autonomous second-order Hamiltonian systems, Electron. J. Differ. Eq., 2013 (2013), 1-13. doi: 10.1186/1687-1847-2013-1

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)