Sandwich pairs in critical point theory
HTML articles powered by AMS MathViewer
- by Martin Schechter PDF
- Trans. Amer. Math. Soc. 360 (2008), 2811-2823 Request permission
Abstract:
Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are critical points of functionals. If a functional $G$ is semibounded, one can find a Palais-Smale (PS) sequence \[ G(u_k) \to a,\quad G’(u_k)\to 0. \] These sequences produce critical points if they have convergent subsequences (i.e., if $G$ satisfies the PS condition). However, there is no clear method of finding critical points of functionals which are not semibounded. The concept of linking was developed to produce Palais-Smale (PS) sequences for $C^1$ functionals $G$ that separate linking sets. In the present paper we discuss the situation in which one cannot find linking sets that separate the functional. We introduce a new class of subsets that accomplishes the same results under weaker conditions. We then provide criteria for determining such subsets. Examples and applications are given.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Marcelo F. Furtado, Liliane A. Maia, and Elves A. B. Silva, On a double resonant problem in $\Bbb R^N$, Differential Integral Equations 15 (2002), no. 11, 1335–1344. MR 1920690
- Marcelo F. Furtado and Elves A. B. Silva, Double resonant problems which are locally non-quadratic at infinity, Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000) Electron. J. Differ. Equ. Conf., vol. 6, Southwest Texas State Univ., San Marcos, TX, 2001, pp. 155–171. MR 1804772
- Martin Schechter, New saddle point theorems, Generalized functions and their applications (Varanasi, 1991) Plenum, New York, 1993, pp. 213–219. MR 1240078, DOI 10.1007/978-1-4899-1591-7_{2}0
- Martin Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (1992), no. 3, 269–281. MR 1201854, DOI 10.4064/ap-57-3-269-281
- Martin Schechter, New linking theorems, Rend. Sem. Mat. Univ. Padova 99 (1998), 255–269. MR 1636619
- Martin Schechter, Linking methods in critical point theory, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1729208, DOI 10.1007/978-1-4612-1596-7
- Elves Alves de B. e Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991), no. 5, 455–477. MR 1093380, DOI 10.1016/0362-546X(91)90070-H
Additional Information
- Martin Schechter
- Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
- Email: mschecht@math.uci.edu
- Received by editor(s): August 14, 2005
- Published electronically: January 25, 2008
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 2811-2823
- MSC (2000): Primary 35J65, 58E05, 49J35
- DOI: https://doi.org/10.1090/S0002-9947-08-04470-X
- MathSciNet review: 2379776