A compression type mountain pass theorem in conical shells


In this paper we present a compression type version of the mountain pass lemma in a conical shell with respect to two norms. An application to second-order ordinary differential equations is included.


Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


Critical point; Mountain pass lemma; Compression; Cone; Positive solution

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R. Precup, A compression type mountain pass theorem in conical shells, J. Math. Anal. Appl. 338 (2008), 1116-1130, https://doi.org/10.1016/j.jmaa.2007.06.007


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[1] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.
[2] H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991) 939–964.
[3] K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.
[4] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[5] M. Frigon, On a new notion of linking and application to elliptic problems at resonance, J. Differential Equations 153 (1999) 96–120.
[6] L. Gasinski, N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman & Hall/CRC, 2005.
[7] A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
[8] 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.
[9] D. Guo, J. Sun, G. Qi, Some extensions of the mountain pass lemma, Differential Integral Equations 1 (1988) 351–358.
[10] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Springer, Paris, 1993.
[11] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
[12] Z. Liu, J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations 172 (2001) 257–299.
[13] L. Ma, Mountain pass on a closed convex set, J. Math. Anal. Appl. 205 (1997) 531–536.
[14] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
[15] D. Motreanu, V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems, Kluwer, Dordrecht, 2003.
[16] D. O’Regan, R. Precup, Theorems of Leray–Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.
[17] R. Precup, On the Palais–Smale condition for Hammerstein integral equations in Hilbert spaces, Nonlinear Anal. 47 (2001) 1233–1244.
[18] R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.
[19] P. Pucci, J. Serrin, A mountain pass theorem, J. Differential Equations 60 (1985) 142–149.
[20] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
[21] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331 (1992) 681–703.
[22] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Basel, 1999.
[23] M. Schechter, K. Tintarev, Nonlinear eigenvalues and mountain pass methods, Topol. Methods Nonlinear Anal. 1 (1993) 183–201.
[24] M. Struwe, Variational Methods, Springer, Berlin, 1990.
[25] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.


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