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On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on N

Published online by Cambridge University Press:  14 November 2011

Louis Jeanjean
Affiliation:
Université de Marne-La-Vallée, Equipe d'Analyse et de Mathématiques Appliquées, 5, bd Descartes, Champs-sur-Marne, 77454 Marne-La-Vallée Cedex 2, France (jeanjean@math.univ-mlv.fr)
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Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form

We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. xN.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1999

References

1Ambrosetti, A.. Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro. Atti Ace. Naz. Lincei 52 (1972), 660667.Google Scholar
2Ambrosetti, A. and Bertotti, M. L.. Homoclinics for second order convervative systems. Partial differential equations and related subjects (ed Miranda, M.). Pitman Research Notes in Mathematics Series (1992).Google Scholar
3Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Fund. Analysis 14 (1973), 349381.CrossRefGoogle Scholar
4Ambrosetti, A. and Struwe, M.. Existence of steady vortex rings in an ideal fluid. Arch. Ration. Mech. Analysis 108 (1989), 97109.CrossRefGoogle Scholar
5Bahri, A. and Lions, P. L.. Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. 45 (1992), 12051215.CrossRefGoogle Scholar
6Bartolo, P., Benci, V. and Fortunate, D.Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity. Nonlinear Analysis 7 (1983), 9811012.CrossRefGoogle Scholar
7Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations I. Arch. Ration. Mech. Analysis 82 (1983), 313346.CrossRefGoogle Scholar
8Brezis, H.. Analyse fonctionnelle (Paris: Masson, 1983).Google Scholar
9Brezis, H. and Nirenberg, L.. Nonlinear Analysis. (In preparation.)Google Scholar
10Buffoni, B., Jeanjean, L. and Stuart, C. A.. Existence of a non-trivial solution to a strongly indefinite semilinear equation. Proc. AMS 119 (1993), 179186.CrossRefGoogle Scholar
11Cerami, G.. Un criterio di esistenza per i punti critici su varieta ilimitate. Rend. Acad. Sci. Let. 1st. Lombardo 112 (1978), 332336.Google Scholar
12Zelati, V. Coti and Rabinowitz, P. H.. Homoclinic type solutions for a semilinear elliptic PDE on N. Commun. Pure Appl. Math. 45 (1992), 12171269.Google Scholar
13Ekeland, I.. On the variational principle. J. Math. Analysis Applic. 47 (1974), 324353.CrossRefGoogle Scholar
14Ghoussoub, N.. Duality and perturbation methods in critical point theory, 107 (Cambridge University Press, 1993).CrossRefGoogle Scholar
15Ginzburg, V. L.. An embedding S2n–12n, 2n – 1 ≥ 7, whose Hamiltonian flow has no periodic trajectories. IMRN 2 (1995), 8398.CrossRefGoogle Scholar
16Herman, M.. Examples of compact hypersurfaces in 2p, 2p ≥ 6, with no periodic orbits. Fax to H. Hofer, 7 December 1994.Google Scholar
17Jeanjean, L.. Solution in spectral gaps for a nonlinear equation of Schrödinger type. J. Diff. Eqns 112 (1994), 5380.CrossRefGoogle Scholar
18Jeanjean, L.. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Analysis 28 (1997), 16331659.CrossRefGoogle Scholar
19Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case. Parts I and II. Ann. Inst. H. Poincaré Analyse non linéaire 1 (1984), 109145; 223–283.CrossRefGoogle Scholar
20Lions, P. L.. Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987), 3397.CrossRefGoogle Scholar
21Rabinowitz, P. H. and Tanaka, K.. Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206 (1991), 473499.CrossRefGoogle Scholar
22Schechter, M. and Tintarev, K.. Spherical maxima in Hilbert space and semilinear elliptic eigenvalue problems. Diff. Int. Eqns 3 (1990), 889899.Google Scholar
23Struwe, M.. Variational methods, 2nd edn (New York: Springer, 1996).CrossRefGoogle Scholar
24Struwe, M.. The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160 (1988), 1964.CrossRefGoogle Scholar
25Struwe, M.. Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Boletim Soc. Bras. Mat. 20 (1990), 4958.CrossRefGoogle Scholar
26Struwe, M.. Une estimation asymptotique pour le modèle Ginzburg–Landau. C. R. Acad. Sci. Pans 317 (1993), 677680.Google Scholar
27Struwe, M. and Tarantello, G.. On multivortex solutions in Chern–Simons gauge theory. Bolletino UMI Nuova serie Sezione Scientifica I-vol (1997).Google Scholar
28Stuart, C. A.. Bifurcation for Dirichlet problems without eigenvalues. Proc. Lond. Math. Soc. 45 (1982), 149162.Google Scholar
29Stuart, C. A.. Bifurcation in LP(N) for a seniilinear elliptic equation. Proc. Lond. Math. Soc. 57 (1988), 511541.CrossRefGoogle Scholar
30Stuart, C. A. and Zhou, H. S.. A variational problem related to self-trapping of an electromagnetic field. Math. Meth. Appl. Sci. 19 (1996), 13971407.3.0.CO;2-B>CrossRefGoogle Scholar
31Tarantello, G.. Nodal solutions of seniilinear elliptic equations with critical exponent. C. R. Acad. Sci. Paris 313 (1991), 441445.Google Scholar
32Zhou, H. S.. Positive solution for a semilinear elliptic equation which is almost linear at infinity. ZAMP 49 (1998), 896906.CrossRefGoogle Scholar
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