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

  • Louis Jeanjean (a1)

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.

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1Ambrosetti A.. Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro. Atti Ace. Naz. Lincei 52 (1972), 660667.
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).
3Ambrosetti A. and Rabinowitz P. H.. Dual variational methods in critical point theory and applications. J. Fund. Analysis 14 (1973), 349381.
4Ambrosetti A. and Struwe M.. Existence of steady vortex rings in an ideal fluid. Arch. Ration. Mech. Analysis 108 (1989), 97109.
5Bahri A. and Lions P. L.. Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. 45 (1992), 12051215.
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.
7Berestycki H. and Lions P. L.. Nonlinear scalar field equations I. Arch. Ration. Mech. Analysis 82 (1983), 313346.
8Brezis H.. Analyse fonctionnelle (Paris: Masson, 1983).
9Brezis H. and Nirenberg L.. Nonlinear Analysis. (In preparation.)
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.
11Cerami G.. Un criterio di esistenza per i punti critici su varieta ilimitate. Rend. Acad. Sci. Let. 1st. Lombardo 112 (1978), 332336.
12Zelati V. Coti and Rabinowitz P. H.. Homoclinic type solutions for a semilinear elliptic PDE on N. Commun. Pure Appl. Math. 45 (1992), 12171269.
13Ekeland I.. On the variational principle. J. Math. Analysis Applic. 47 (1974), 324353.
14Ghoussoub N.. Duality and perturbation methods in critical point theory, 107 (Cambridge University Press, 1993).
15Ginzburg V. L.. An embedding S2n–12n, 2n – 1 ≥ 7, whose Hamiltonian flow has no periodic trajectories. IMRN 2 (1995), 8398.
16Herman M.. Examples of compact hypersurfaces in 2p, 2p ≥ 6, with no periodic orbits. Fax to H. Hofer, 7 December 1994.
17Jeanjean L.. Solution in spectral gaps for a nonlinear equation of Schrödinger type. J. Diff. Eqns 112 (1994), 5380.
18Jeanjean L.. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Analysis 28 (1997), 16331659.
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.
20Lions P. L.. Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987), 3397.
21Rabinowitz P. H. and Tanaka K.. Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206 (1991), 473499.
22Schechter M. and Tintarev K.. Spherical maxima in Hilbert space and semilinear elliptic eigenvalue problems. Diff. Int. Eqns 3 (1990), 889899.
23Struwe M.. Variational methods, 2nd edn (New York: Springer, 1996).
24Struwe M.. The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160 (1988), 1964.
25Struwe M.. Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Boletim Soc. Bras. Mat. 20 (1990), 4958.
26Struwe M.. Une estimation asymptotique pour le modèle Ginzburg–Landau. C. R. Acad. Sci. Pans 317 (1993), 677680.
27Struwe M. and Tarantello G.. On multivortex solutions in Chern–Simons gauge theory. Bolletino UMI Nuova serie Sezione Scientifica I-vol (1997).
28Stuart C. A.. Bifurcation for Dirichlet problems without eigenvalues. Proc. Lond. Math. Soc. 45 (1982), 149162.
29Stuart C. A.. Bifurcation in LP(N) for a seniilinear elliptic equation. Proc. Lond. Math. Soc. 57 (1988), 511541.
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.
31Tarantello G.. Nodal solutions of seniilinear elliptic equations with critical exponent. C. R. Acad. Sci. Paris 313 (1991), 441445.
32Zhou H. S.. Positive solution for a semilinear elliptic equation which is almost linear at infinity. ZAMP 49 (1998), 896906.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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