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Non-smooth critical point theory on closed convex sets

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  • 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.
    Mathematics Subject Classification: Primary: 58E05, 49J35; Secondary: 49J52.

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  • [1]

    K. Borsuk, Theory of Retracts, PWN, Warsaw, 1967.

    [2]

    H. Brézis and L. Nirenberg, Remarks on finding critical points, Comm. Pure. Appl. Math., 44 (1991), 939-963.doi: 10.1002/cpa.3160440808.

    [3]

    K.-C. Chang, Variational methods for nondifferentiable functions and their applications to partial differential equations , J. Math. Anal. Appl., 80 (1981), 102-129.doi: 10.1016/0022-247X(81)90095-0.

    [4]

    K.-C. Chang, On the mountain pass lemma , in Equadiff 6 (Brno, 1985), Lecture Notes in Math., 1192, Springer, Berlin, (1986), 203-208.doi: 10.1007/BFb0076070.

    [5]

    K.-C. Chang and J. Eells, Unstable minimal surface coboundaries , Acta Math. Sin. (Engl. Ser.), 2 (1986), 233-247.doi: 10.1007/BF02582026.

    [6]

    J. Chen, Some new generalizations of critical point theorems for locally Lipschitz functions , J. Appl. Anal., 14 (2008), 193-208.doi: 10.1515/JAA.2008.193.

    [7]

    M. Choulli, R. Deville and A. Rhandi, A general mountain pass principle for nondifferentiable functionals and applications , Rev. Mat. Apl., 13 (1992), 45-58.

    [8]

    F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math., 5, SIAM, Philadelphia, 1990.doi: 10.1137/1.9781611971309.

    [9]

    L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.

    [10]

    N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Math., 107, Cambridge Univ. Press, Cambridge, 1993.doi: 10.1017/CBO9780511551703.

    [11]

    A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications , Nonlinear Anal., 72 (2010), 1319-1338.doi: 10.1016/j.na.2009.08.001.

    [12]

    Y. Jabri, The Mountain Pass Theorem: Variants, Generalizations and some Applications, Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge, 2003.doi: 10.1017/CBO9780511546655.

    [13]

    N. C. Kourogenis and N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance , J. Austral. Math. Soc. Ser. A, 69 (2000), 245-271.

    [14]

    S. Th. Kyritsi and 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.

    [15]

    S. Th. Kyritsi and N. S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities , Nonlinear Anal., 61 (2005), 373-403.doi: 10.1016/j.na.2004.12.001.

    [16]

    R. Livrea and S. A. Marano, Existence and classification of critical points for non-differentiable functions , Adv. Differential Equations, 9 (2004), 961-978.

    [17]

    R. Livrea and S. A. Marano, Non-smooth critical point theory , in Handbook of Nonconvex Analysis and Applications (eds. D. Y. Gao and D. Motreanu), International Press, (2010), 353-408.

    [18]

    L. Ma, Mountain Pass on a closed convex set , J. Math. Anal. Appl., 205 (1997), 531-536.doi: 10.1006/jmaa.1997.5227.

    [19]

    S. A. Marano and D. Motreanu, Critical points of non-smooth functions with a weak compactness condition , J. Math. Anal. Appl., 358 (2009), 189-201.doi: 10.1016/j.jmaa.2009.04.056.

    [20]

    E. Michael, Continuous selections. I , Ann. of Math., 63 (1956), 361-382.

    [21]

    D. Motreanu, V. V. Motreanu and D. Pasca, A version of Zhong's coercivity result for a general class of nonsmooth functionals , Abst. Appl. Anal., 7 (2002), 601-612.doi: 10.1155/S1085337502207058.

    [22]

    D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optim. Appl., 29, Kluwer, Dordrecht, 1998.

    [23]

    D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optim. Appl., 67, Kluwer, Dordrecht, 2003.

    [24]

    V. D. Radulescu, Mountain pass theorems for non-differentiable functions and applications , Proc. Japan Acad., 69 (1993), 193-198.

    [25]

    M. Sion, On general minimax theorems , Pacific J. Math., 8 (1958), 171-176.

    [26]

    M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Ergeb. Math. Grenzgeb, 34, Springer-Verlag, Berlin, 1996.

    [27]

    A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems , Ann. Inst. Henri Poincaré, \textbf{3} (1986), 77-109.

    [28]

    C. Zhong, On Ekeland's variational principle and a minimax theorem , J. Math. Anal. Appl., 205 (1997), 239-250.doi: 10.1006/jmaa.1996.5168.

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