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May  2014, 13(3): 1187-1202. doi: 10.3934/cpaa.2014.13.1187

Non-smooth critical point theory on closed convex sets

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania

Received  May 2013 Revised  September 2013 Published  December 2013

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.
Citation: Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
References:
[1]

K. Borsuk, Theory of Retracts,, PWN, (1967).   Google Scholar

[2]

H. Brézis and L. Nirenberg, Remarks on finding critical points,, \emph{Comm. Pure. Appl. Math.}, 44 (1991), 939.  doi: 10.1002/cpa.3160440808.  Google Scholar

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

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K.-C. Chang, On the mountain pass lemma ,, in \emph{Equadiff 6 (Brno, 1192 (1986), 203.  doi: 10.1007/BFb0076070.  Google Scholar

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

[6]

J. Chen, Some new generalizations of critical point theorems for locally Lipschitz functions ,, \emph{J. Appl. Anal.}, 14 (2008), 193.  doi: 10.1515/JAA.2008.193.  Google Scholar

[7]

M. Choulli, R. Deville and A. Rhandi, A general mountain pass principle for nondifferentiable functionals and applications ,, \emph{Rev. Mat. Apl.}, 13 (1992), 45.   Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Classics Appl. Math., 5 (1990).  doi: 10.1137/1.9781611971309.  Google Scholar

[9]

L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Ser. Math. Anal. Appl., 8 (2005).   Google Scholar

[10]

N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory,, Cambridge Tracts in Math., 107 (1993).  doi: 10.1017/CBO9780511551703.  Google Scholar

[11]

A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications ,, \emph{Nonlinear Anal.}, 72 (2010), 1319.  doi: 10.1016/j.na.2009.08.001.  Google Scholar

[12]

Y. Jabri, The Mountain Pass Theorem: Variants, Generalizations and some Applications,, Encyclopedia Math. Appl., (2003).  doi: 10.1017/CBO9780511546655.  Google Scholar

[13]

N. C. Kourogenis and N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance ,, \emph{J. Austral. Math. Soc. Ser. A}, 69 (2000), 245.   Google Scholar

[14]

S. Th. Kyritsi and N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance ,, \emph{J. Math. Anal. Appl.}, 276 (2002), 292.  doi: 10.1016/S0022-247X(02)00443-2.  Google Scholar

[15]

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

[16]

R. Livrea and S. A. Marano, Existence and classification of critical points for non-differentiable functions ,, \emph{Adv. Differential Equations}, 9 (2004), 961.   Google Scholar

[17]

R. Livrea and S. A. Marano, Non-smooth critical point theory ,, in \emph{Handbook of Nonconvex Analysis and Applications} (eds. D. Y. Gao and D. Motreanu), (2010), 353.   Google Scholar

[18]

L. Ma, Mountain Pass on a closed convex set ,, \emph{J. Math. Anal. Appl.}, 205 (1997), 531.  doi: 10.1006/jmaa.1997.5227.  Google Scholar

[19]

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

[20]

E. Michael, Continuous selections. I ,, \emph{Ann. of Math.}, 63 (1956), 361.   Google Scholar

[21]

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

[22]

D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,, \emph{Nonconvex Optim. Appl.}, 29 (1998).   Google Scholar

[23]

D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems,, Nonconvex Optim. Appl., 67 (2003).   Google Scholar

[24]

V. D. Radulescu, Mountain pass theorems for non-differentiable functions and applications ,, \emph{Proc. Japan Acad.}, 69 (1993), 193.   Google Scholar

[25]

M. Sion, On general minimax theorems ,, \emph{Pacific J. Math.}, 8 (1958), 171.   Google Scholar

[26]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition,, Ergeb. Math. Grenzgeb, 34 (1996).   Google Scholar

[27]

A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems ,, \emph{Ann. Inst. Henri Poincar\'{e, 3 (1986), 77.   Google Scholar

[28]

C. Zhong, On Ekeland's variational principle and a minimax theorem ,, \emph{J. Math. Anal. Appl.}, 205 (1997), 239.  doi: 10.1006/jmaa.1996.5168.  Google Scholar

show all references

References:
[1]

K. Borsuk, Theory of Retracts,, PWN, (1967).   Google Scholar

[2]

H. Brézis and L. Nirenberg, Remarks on finding critical points,, \emph{Comm. Pure. Appl. Math.}, 44 (1991), 939.  doi: 10.1002/cpa.3160440808.  Google Scholar

[3]

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

[4]

K.-C. Chang, On the mountain pass lemma ,, in \emph{Equadiff 6 (Brno, 1192 (1986), 203.  doi: 10.1007/BFb0076070.  Google Scholar

[5]

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

[6]

J. Chen, Some new generalizations of critical point theorems for locally Lipschitz functions ,, \emph{J. Appl. Anal.}, 14 (2008), 193.  doi: 10.1515/JAA.2008.193.  Google Scholar

[7]

M. Choulli, R. Deville and A. Rhandi, A general mountain pass principle for nondifferentiable functionals and applications ,, \emph{Rev. Mat. Apl.}, 13 (1992), 45.   Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Classics Appl. Math., 5 (1990).  doi: 10.1137/1.9781611971309.  Google Scholar

[9]

L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,, Ser. Math. Anal. Appl., 8 (2005).   Google Scholar

[10]

N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory,, Cambridge Tracts in Math., 107 (1993).  doi: 10.1017/CBO9780511551703.  Google Scholar

[11]

A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications ,, \emph{Nonlinear Anal.}, 72 (2010), 1319.  doi: 10.1016/j.na.2009.08.001.  Google Scholar

[12]

Y. Jabri, The Mountain Pass Theorem: Variants, Generalizations and some Applications,, Encyclopedia Math. Appl., (2003).  doi: 10.1017/CBO9780511546655.  Google Scholar

[13]

N. C. Kourogenis and N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance ,, \emph{J. Austral. Math. Soc. Ser. A}, 69 (2000), 245.   Google Scholar

[14]

S. Th. Kyritsi and N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance ,, \emph{J. Math. Anal. Appl.}, 276 (2002), 292.  doi: 10.1016/S0022-247X(02)00443-2.  Google Scholar

[15]

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

[16]

R. Livrea and S. A. Marano, Existence and classification of critical points for non-differentiable functions ,, \emph{Adv. Differential Equations}, 9 (2004), 961.   Google Scholar

[17]

R. Livrea and S. A. Marano, Non-smooth critical point theory ,, in \emph{Handbook of Nonconvex Analysis and Applications} (eds. D. Y. Gao and D. Motreanu), (2010), 353.   Google Scholar

[18]

L. Ma, Mountain Pass on a closed convex set ,, \emph{J. Math. Anal. Appl.}, 205 (1997), 531.  doi: 10.1006/jmaa.1997.5227.  Google Scholar

[19]

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

[20]

E. Michael, Continuous selections. I ,, \emph{Ann. of Math.}, 63 (1956), 361.   Google Scholar

[21]

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

[22]

D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,, \emph{Nonconvex Optim. Appl.}, 29 (1998).   Google Scholar

[23]

D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems,, Nonconvex Optim. Appl., 67 (2003).   Google Scholar

[24]

V. D. Radulescu, Mountain pass theorems for non-differentiable functions and applications ,, \emph{Proc. Japan Acad.}, 69 (1993), 193.   Google Scholar

[25]

M. Sion, On general minimax theorems ,, \emph{Pacific J. Math.}, 8 (1958), 171.   Google Scholar

[26]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition,, Ergeb. Math. Grenzgeb, 34 (1996).   Google Scholar

[27]

A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems ,, \emph{Ann. Inst. Henri Poincar\'{e, 3 (1986), 77.   Google Scholar

[28]

C. Zhong, On Ekeland's variational principle and a minimax theorem ,, \emph{J. Math. Anal. Appl.}, 205 (1997), 239.  doi: 10.1006/jmaa.1996.5168.  Google Scholar

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