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

 Access to full text

 Full (PDF)

 Full (DjVu)

1. AMBROSETTI, A. - MALCHIODI, A., Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007. Zbl1125.47052MR2292344DOI10.1017/CBO9780511618260
2. AMBROSETTI, A. - RABINOWITZ, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Zbl0273.49063MR370183
3. BREZIS, H. - BROWDER, F., Partial differential equations in the 20th century, Adv. Math., 135 (1998), 76-144. Zbl0915.01011MR1617413DOI10.1006/aima.1997.1713
4. BREZIS, H. - CORON, J.-M. - NIRENBERG, L., Free vibrations for a nonlinear wave equation and a theorem of Rabinowitz, Comm. Pure Appl. Math., 33 (1980), 667-689. Zbl0484.35057MR586417DOI10.1002/cpa.3160330507
5. BREZIS, H. - NIRENBERG, L., Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-964. Zbl0751.58006MR1127041DOI10.1002/cpa.3160440808
6. CERAMI, G., Un criterio di esistenza per i punti critici su varietà illimitate, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336. MR581298
7. CHANG, K. C., Variational methods for non-differentiable functionals and applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. Zbl0487.49027MR614246DOI10.1016/0022-247X(81)90095-0
8. CLARKE, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262. Zbl0307.26012MR367131DOI10.2307/1997202
9. CLARKE, F. H., Solution périodique des équations hamiltoniennes, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A951-A952. Zbl0422.35005MR520777
10. CLARKE, F. H., Generalized gradients of Lipschitz functionals, Adv. in Math., 40 (1981), 52-67. Zbl0463.49017MR616160DOI10.1016/0001-8708(81)90032-3
11. DUGUNDJI, J., Topology, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1966. MR193606
12. EKELAND, I., On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. Zbl0286.49015MR346619DOI10.1016/0022-247X(74)90025-0
13. EKELAND, I., Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474. Zbl0441.49011MR526967DOI10.1090/S0273-0979-1979-14595-6
14. EMDEN, R., Die GaskuÈgeln, Teubner, Leipzig, 1907. 
15. FOWLER, R. H., Further studies of Emden and similar differential equations, Quart. J. Math., 2 (1931), 259-288. Zbl57.0523.02
16. GHERGU, M. - RĂDULESCU, V., Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford University Press, 2008. MR2488149
17. GHOUSSOUB, N. - PREISS, D., A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 6 (1989), 321-330. Zbl0711.58008MR1030853
18. JABRI, Y., The Mountain Pass Theorem. Variants, Generalizations and Some Applications, Encyclopedia of Mathematics and its Applications, vol. 95, Cambridge University Press, Cambridge, 2003. Zbl1036.49001MR2012778DOI10.1017/CBO9780511546655
19. KAZDAN, J. L. - WARNER, F. W., Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom., 10 (1975), 113-134. Zbl0296.53037MR365409
20. KAZDAN, J. L. - WARNER, F. W., Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597. Zbl0325.35038MR477445DOI10.1002/cpa.3160280502
21. KRISTÁLY, A. - RĂDULESCU, V. - VARGA, CS., 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. MR2683404DOI10.1017/CBO9780511760631
22. LEBOURG, G., Valeur moyenne pour gradient generalisé, C. R. Acad. Sci. Paris, 281 (1975), 795-797. Zbl0317.46034MR388097
23. LI, S., Some aspects of nonlinear operators and critical point theory, Functional analysis in China ( Li, Bingren, Eds.), Kluwer Academic Publishers, Dordrecht, 1996, 132-144. Zbl0847.58013MR1379601
24. MAWHIN, J. - WILLEM, M., Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74, Springer, New York, 1989. Zbl0676.58017MR982267DOI10.1007/978-1-4757-2061-7
25. PALAIS, R., Ljusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132. Zbl0143.35203MR259955DOI10.1016/0040-9383(66)90013-9
26. PALAIS, R. - SMALE, S., A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165-171. MR158411DOI10.1090/S0002-9904-1964-11062-4
27. POHOZAEV, S., On the eigenfunctions of the equation $\mathrm{\Delta }u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. MR192184
28. PUCCI, P. - SERRIN, J., Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. Zbl0564.58012MR766489DOI10.1016/0022-1236(84)90072-7
29. PUCCI, P. - SERRIN, J., A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. Zbl0585.58006MR808262DOI10.1016/0022-0396(85)90125-1
30. PUCCI, P. - SERRIN, J., A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703. Zbl0625.35027MR855181DOI10.1512/iumj.1986.35.35036
31. PUCCI, P. - SERRIN, J., The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc., 299 (1987), 115-132. Zbl0611.58019MR869402DOI10.2307/2000484
32. PUCCI, P. - SERRIN, J., A note on the strong maximum principle for elliptic differential inequalities, J. Math. Pures Appl., 79 (2000), 57-71. Zbl0952.35163MR1742565DOI10.1016/S0021-7824(99)00146-4
33. PUCCI, P. - SERRIN, J., The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66; Erratum, J. Differential Equations, 207 (2004), 226-227. Zbl1109.35022MR2100819DOI10.1016/j.jde.2004.09.002
34. PUCCI, P. - SERRIN, J., 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. Zbl1193.35024MR2569337
35. RABINOWITZ, P., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conference Series in Mathematics, 65, 1986American Mathematical Society, Providence, RI. MR845785DOI10.1090/cbms/065
36. RĂDULESCU, V., Mountain pass theorems for non-differentiable functions and applications, Proc. Japan Acad., 69A (1993), 193-198. MR1232824
37. RELLICH, F., Darstellung der Eigenwerte von $\mathrm{\Delta }u+\lambda u=0$ durch ein Randintegral, Math. Z., 46 (1940), 635-636. MR2456DOI10.1007/BF01181459
38. RUDIN, W., Functional Analysis, Mc Graw-Hill, 1973. MR365062
39. SCHECHTER, M., Minimax Systems and Critical Point Theory, Birkhäuser, Boston, 2009. Zbl1186.35043MR2512303DOI10.1007/978-0-8176-4902-9
40. SERRIN, J., Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302. Zbl0128.09101MR170096DOI10.1007/BF02391014
41. STRUWE, M., Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. MR1078018DOI10.1007/978-3-662-02624-3
42. WILLEM, M., Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, 1996. MR1400007DOI10.1007/978-1-4612-4146-1
43. ZOU, W., Sign-changing Critical Point Theory, Springer, New York, 2008. Zbl1159.49010MR2442765