Pucci, Rădulescu: The Impact of the Mountain Pass Theory in Nonlinear Analysis: a Mathematical Survey

Pucci, Patrizia and Rădulescu, Vicenṭiu:
The Impact of the Mountain Pass Theory in Nonlinear Analysis: a Mathematical Survey
Bollettino dell'Unione Matematica Italiana Serie 9 3 (2010), fasc. n.3, p. 543-582, (English)
pdf, djvu. | MR 2742781 | Zbl 1225.49004

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We provide a survey on the mountain pass theory, viewed as a central tool in the modern nonlinear analysis. The abstract results are illustrated with relevant applications to nonlinear partial differential equations.

Referenze Bibliografiche

[1] A. AMBROSETTI - A. MALCHIODI, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007. | fulltext (doi) | MR 2292344 | Zbl 1125.47052

[2] A. AMBROSETTI - P. RABINOWITZ, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. | MR 370183 | Zbl 0273.49063

[3] H. BREZIS - F. BROWDER, Partial differential equations in the 20th century, Adv. Math., 135 (1998), 76-144. | fulltext (doi) | MR 1617413 | Zbl 0915.01011

[4] H. BREZIS - J.-M. CORON - L. NIRENBERG, Free vibrations for a nonlinear wave equation and a theorem of Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667-689. | fulltext (doi) | MR 586417 | Zbl 0484.35057

[5] H. BREZIS - L. NIRENBERG, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-964. | fulltext (doi) | MR 1127041 | Zbl 0751.58006

[6] G. CERAMI, Un criterio di esistenza per i punti critici su varietà illimitate, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336. | MR 581298

[7] K. C. CHANG, Variational methods for non-differentiable functionals and applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. | fulltext (doi) | MR 614246 | Zbl 0487.49027

[8] F. H. CLARKE, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262. | fulltext (doi) | MR 367131 | Zbl 0307.26012

[9] F. H. CLARKE, Solution périodique des équations hamiltoniennes, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A951-A952. | MR 520777 | Zbl 0422.35005

[10] F. H. CLARKE, Generalized gradients of Lipschitz functionals, Adv. in Math., 40 (1981), 52-67. | fulltext (doi) | MR 616160 | Zbl 0463.49017

[11] J. DUGUNDJI, Topology, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1966. | MR 193606

[12] I. EKELAND, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. | fulltext (doi) | MR 346619 | Zbl 0286.49015

[13] I. EKELAND, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474. | fulltext (doi) | MR 526967 | Zbl 0441.49011

[14] R. EMDEN, Die GaskuÈgeln, Teubner, Leipzig, 1907.

[15] R. H. FOWLER, Further studies of Emden and similar differential equations, Quart. J. Math., 2 (1931), 259-288. | Zbl 57.0523.02

[16] M. GHERGU - V. RĂDULESCU, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford University Press, 2008. | MR 2488149

[17] N. GHOUSSOUB - D. PREISS, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 6 (1989), 321-330. | fulltext EuDML | MR 1030853 | Zbl 0711.58008

[18] Y. JABRI, The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Encyclopedia of Mathematics and its Applications, vol. 95, Cambridge University Press, Cambridge, 2003. | fulltext (doi) | MR 2012778 | Zbl 1036.49001

[19] J. L. KAZDAN - F. W. WARNER, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom., 10 (1975), 113-134. | MR 365409 | Zbl 0296.53037

[20] J. L. KAZDAN - F. W. WARNER, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597. | fulltext (doi) | MR 477445 | Zbl 0325.35038

[21] A. KRISTÁLY - V. RĂDULESCU - CS. VARGA, Variational Principles in Mathematical Physics, Geometry, and Economics. Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, vol. 136, Cambridge University Press, Cambridge, 2010. | fulltext (doi) | MR 2683404

[22] G. LEBOURG, Valeur moyenne pour gradient generalisé, C. R. Acad. Sci. Paris, 281 (1975), 795-797. | MR 388097 | Zbl 0317.46034

[23] S. LI, Some aspects of nonlinear operators and critical point theory, Functional analysis in China (Li, Bingren, Eds.), Kluwer Academic Publishers, Dordrecht, 1996, 132-144. | MR 1379601 | Zbl 0847.58013

[24] J. MAWHIN - M. WILLEM, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74, Springer, New York, 1989. | fulltext (doi) | MR 982267 | Zbl 0676.58017

[25] R. PALAIS, Ljusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132. | fulltext (doi) | MR 259955 | Zbl 0143.35203

[26] R. PALAIS - S. SMALE, A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165-171. | fulltext (doi) | MR 158411

[27] S. POHOZAEV, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. | MR 192184

[28] P. PUCCI - J. SERRIN, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. | fulltext (doi) | MR 766489 | Zbl 0564.58012

[29] P. PUCCI - J. SERRIN, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. | fulltext (doi) | MR 808262 | Zbl 0585.58006

[30] P. PUCCI - J. SERRIN, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. | fulltext (doi) | MR 855181 | Zbl 0625.35027

[31] P. PUCCI - J. SERRIN, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc., 299 (1987), 115-132. | fulltext (doi) | MR 869402 | Zbl 0611.58019

[32] P. PUCCI - J. SERRIN, A note on the strong maximum principle for elliptic differential inequalities, J. Math. Pures Appl., 79 (2000), 57-71. | fulltext (doi) | MR 1742565 | Zbl 0952.35163

[33] P. PUCCI - J. SERRIN, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66; Erratum, J. Differential Equations, 207 (2004), 226-227. | fulltext (doi) | MR 2100819 | Zbl 1109.35022

[34] P. PUCCI - J. SERRIN, On the strong maximum and compact support principles and some applications, in Handbook of Differential Equations - Stationary Partial Differential Equations (M. Chipot, Ed.), Elsevier, Amsterdam, Vol. 4 (2007), 355-483. | MR 2569337 | Zbl 1193.35024

[35] P. RABINOWITZ, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conference Series in Mathematics, 65, 1986 American Mathematical Society, Providence, RI. | fulltext (doi) | MR 845785

[36] V. RĂDULESCU, Mountain pass theorems for non-differentiable functions and applications, Proc. Japan Acad., 69A (1993), 193-198. | MR 1232824

[37] F. RELLICH, Darstellung der Eigenwerte von $\Delta u + \lambda u = 0$ durch ein Randintegral, Math. Z., 46 (1940), 635-636. | fulltext EuDML | fulltext (doi) | MR 2456

[38] W. RUDIN, Functional Analysis, Mc Graw-Hill, 1973. | MR 365062

[39] M. SCHECHTER, Minimax Systems and Critical Point Theory, Birkhäuser, Boston, 2009. | fulltext (doi) | MR 2512303 | Zbl 1186.35043

[40] J. SERRIN, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302. | fulltext (doi) | MR 170096 | Zbl 0128.09101

[41] M. STRUWE, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. | fulltext (doi) | MR 1078018

[42] M. WILLEM, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, 1996. | fulltext (doi) | MR 1400007

[43] W. ZOU, Sign-changing Critical Point Theory, Springer, New York, 2008. | MR 2442765 | Zbl 1159.49010

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Referenze Bibliografiche