zbMATH — the first resource for mathematics

A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P. S. condition and coercivity. (English) Zbl 0912.49021
The paper gives a generalization of the celebrated variational principle formulated by Ekeland in 1974. The main hypotheses are the same, excepting the contribution of a weight function \(h:[0,+\infty)\rightarrow [0,+\infty)\) which models the null map in the sense that \(h\) is continuous, nondecreasing and \(\int_0^\infty (1/(1+h(r))dr=+\infty\). The proof follows inductively and uses the same steps as in the classical case. The abstract result is illustrated by several examples, under the additional assumption that the functional satisfies the weak Palais-Smale condition, in the variant used by Cerami. It is refound the well-known result which asserts that a lower semicontinuous, proper, Gâteaux-differentiable bounded from below functional achieves its minimum, provided that the weak Palais-Smale condition is fulfilled. It is also analysed in the paper the link between this compactness condition and the notion of coercivity.

49K27 Optimality conditions for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E20 Harmonic maps, etc.
Full Text: DOI
[1] Brezis, H.; Nirenberg, L., Remarks on finding critical points, Comm. pure appl. math., 64, 939-963, (1991) · Zbl 0751.58006
[2] Ekeland, I., On the variational principle, J. math. anal. applic., 47, 324-357, (1974) · Zbl 0286.49015
[3] Shi, S.Z., Ekerland’s variational principle and the mountain pass lemma, Acta math. sinca (NS), 1, 348-358, (1985)
[4] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear analysis, 7, 9, 981-1012, (1983) · Zbl 0522.58012
[5] Cerami, G., Un criterio di esistenza per i punti critici su varieta illimitate, Rc. ist. lomb. sci. lett., 112, 332-336, (1978) · Zbl 0436.58006
[6] Caklovic, L.; Li, S.J.; Willem, M., A note on Palais-Smale condition and coercivity, Diff. int. eqn., 3, 799-800, (1990) · Zbl 0782.58019
[7] Costa, D.G.; Elves, de B. e Silva, The Palais-Smale condition versus coercivity, Nonlinear analysis, 16, 371-381, (1991) · Zbl 0733.49011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.