## Abstract and Applied Analysis

### A version of Zhong′s coercivity result for a general class of nonsmooth functionals

#### Abstract

A version of Zhong′s coercivity result (1997) is established for nonsmooth functionals expressed as a sum $\Phi +\Psi$, where $\Phi$ is locally Lipschitz and $\Psi$ is convex, lower semicontinuous, and proper. This is obtained as a consequence of a general result describing the asymptotic behavior of the functions verifying the above structure hypothesis. Our approach relies on a version of Ekeland′s variational principle. In proving our coercivity result we make use of a new general Palais-Smale condition. The relationship with other results is discussed.

#### Article information

Source
Abstr. Appl. Anal., Volume 7, Number 11 (2002), 601-612.

Dates
First available in Project Euclid: 14 April 2003

https://projecteuclid.org/euclid.aaa/1050348308

Digital Object Identifier
doi:10.1155/S1085337502207058

Mathematical Reviews number (MathSciNet)
MR1945448

Zentralblatt MATH identifier
1016.58005

#### Citation

Motreanu, D.; Motreanu, V. V.; Paşca, D. A version of Zhong′s coercivity result for a general class of nonsmooth functionals. Abstr. Appl. Anal. 7 (2002), no. 11, 601--612. doi:10.1155/S1085337502207058. https://projecteuclid.org/euclid.aaa/1050348308

#### References

• H. Brézis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963.
• L. Čaklović, S. J. Li, and M. Willem, A note on Palais-Smale condition and coercivity, Differential Integral Equations 3 (1990), no. 4, 799–800.
• G. Cerami, An existence criterion čommentWe changed reference [3 according to MathSciNet database which is differenet from the original manuscript. Please check.?] for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332–336.
• K.-C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102–129.
• F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1983.
• I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.
• D. Goeleven, A note on Palais-Smale condition in the sense of Szulkin, Differential Integral Equations 6 (1993), no. 5, 1041–1043.
• D. Motreanu and V. V. Motreanu, Coerciveness property for a class of non-smooth functionals, Z. Anal. Anwendungen 19 (2000), no. 4, 1087–1093.
• D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems čommentWe removed “and Applications” from the title of [9 according to MathSciNet database. Please check.?] and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and Its Applications, vol. 29, Kluwer Academic Publishers, Dordrecht, 1999.
• A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 2, 77–109.
• C.-K. Zhong, A generalization of Ekeland's variational principle and application to the study of the relation between the weak P.S. condition and coercivity, Nonlinear Anal. 29 (1997), no. 12, 1421–1431.