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Non-smooth critical point theory on closed convex sets. (English) Zbl 1286.58008
Summary: A critical point theory for non-differentiable functionals defined on a closed convex subset of a Banach space is worked out. Special attention is paid to the notion of critical point and possible compactness conditions of Palais-Smale’s type. Two Mountain-Pass like theorems are also established. Concepts and results are compared with those already existing in the literature.

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49J35 Existence of solutions for minimax problems
49J52 Nonsmooth analysis
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[1] K. Borsuk, <em>Theory of Retracts</em>,, PWN, (1967) · Zbl 0153.52905
[2] H. Br\'ezis, <em>Remarks on finding critical points</em>,, Comm. Pure. Appl. Math., 44, 939, (1991) · Zbl 0751.58006
[3] K.-C. Chang, Variational methods for nondifferentiable functions and their applications to partial differential equations ,, J. Math. Anal. Appl., 80, 102, (1981) · Zbl 0487.49027
[4] K.-C. Chang, On the mountain pass lemma ,, in Equadiff 6 (Brno, 1192, 203, (1986) · Zbl 0604.58015
[5] K.-C. Chang, Unstable minimal surface coboundaries ,, Acta Math. Sin. (Engl. Ser.), 2, 233, (1986) · Zbl 0616.58011
[6] J. Chen, Some new generalizations of critical point theorems for locally Lipschitz functions ,, J. Appl. Anal., 14, 193, (2008) · Zbl 1162.58004
[7] M. Choulli, A general mountain pass principle for nondifferentiable functionals and applications ,, Rev. Mat. Apl., 13, 45, (1992) · Zbl 0783.49003
[8] F. H. Clarke, <em>Optimization and Nonsmooth Analysis</em>,, Classics Appl. Math., 5, (1990) · Zbl 0696.49002
[9] L. Gasi\'nski, <em>Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems</em>,, Ser. Math. Anal. Appl., 8, (2005) · Zbl 1058.58005
[10] N. Ghoussoub, <em>Duality and Perturbation Methods in Critical Point Theory</em>,, Cambridge Tracts in Math., 107, (1993) · Zbl 0790.58002
[11] A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications ,, Nonlinear Anal., 72, 1319, (2010) · Zbl 1186.35087
[12] Y. Jabri, <em>The Mountain Pass Theorem: Variants, Generalizations and some Applications</em>,, Encyclopedia Math. Appl., (2003) · Zbl 1036.49001
[13] N. C. Kourogenis, Nonsmooth critical point theory and nonlinear elliptic equations at resonance ,, J. Austral. Math. Soc. Ser. A, 69, 245, (2000) · Zbl 0964.35055
[14] S. Th. Kyritsi, An obstacle problem for nonlinear hemivariational inequalities at resonance ,, J. Math. Anal. Appl., 276, 292, (2002) · Zbl 1022.49015
[15] S. Th. Kyritsi, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities ,, Nonlinear Anal., 61, 373, (2005) · Zbl 1067.49005
[16] R. Livrea, Existence and classification of critical points for non-differentiable functions ,, Adv. Differential Equations, 9, 961, (2004) · Zbl 1100.58008
[17] R. Livrea, Non-smooth critical point theory ,, in Handbook of Nonconvex Analysis and Applications · Zbl 1216.58004
[18] L. Ma, Mountain Pass on a closed convex set ,, J. Math. Anal. Appl., 205, 531, (1997) · Zbl 0895.58055
[19] S. A. Marano, Critical points of non-smooth functions with a weak compactness condition ,, J. Math. Anal. Appl., 358, 189, (2009) · Zbl 1165.49003
[20] E. Michael, Continuous selections. I ,, Ann. of Math., 63, 361, (1956) · Zbl 0071.15902
[21] D. Motreanu, A version of Zhong’s coercivity result for a general class of nonsmooth functionals ,, Abst. Appl. Anal., 7, 601, (2002) · Zbl 1016.58005
[22] D. Motreanu, <em>Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities</em>,, Nonconvex Optim. Appl., 29, (1998)
[23] D. Motreanu, <em>Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems</em>,, Nonconvex Optim. Appl., 67, (2003) · Zbl 1040.49001
[24] V. D. Radulescu, Mountain pass theorems for non-differentiable functions and applications ,, Proc. Japan Acad., 69, 193, (1993) · Zbl 0801.35019
[25] M. Sion, On general minimax theorems ,, Pacific J. Math., 8, 171, (1958) · Zbl 0081.11502
[26] M. Struwe, <em>Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems</em>, Second Edition,, Ergeb. Math. Grenzgeb, 34, (1996)
[27] A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems ,, Ann. Inst. Henri Poincar\'e , 3, 77, (1986) · Zbl 0612.58011
[28] C. Zhong, On Ekeland’s variational principle and a minimax theorem ,, J. Math. Anal. Appl., 205, 239, (1997) · Zbl 0870.49015
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