Topological Methods in Nonlinear Analysis

A topological approach to superlinear indefinite boundary value problems

Duccio Papini and Fabio Zanolin

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We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation $\ddot{x} + q(t) g(x) = 0$, where $g(x)$ has superlinear growth at infinity and $q(t)$ changes sign.

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Topol. Methods Nonlinear Anal., Volume 15, Number 2 (2000), 203-233.

First available in Project Euclid: 22 August 2016

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Papini, Duccio; Zanolin, Fabio. A topological approach to superlinear indefinite boundary value problems. Topol. Methods Nonlinear Anal. 15 (2000), no. 2, 203--233.

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  • R. A. Adams, Sobolev Spaces, Academic Press, New York (1975) \ref\key 2
  • S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking , Ann. Inst. H. Poincaré Anal. Non Linéaire, 13(1996), 95–115 \ref\key 3
  • S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, , Calc. Var. Partial Differential Equations, 1(1993), 439–475 \ref\key 4
  • S. Alama and G. Tarantello, Elliptic problems with nonlinearities indefinite in sign , J. Funct. Anal., 141(1996), 159–215 \ref\key 5
  • H. Amann and J. López–Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems , J. Differential Equations, 146(1998), 336–374 \ref\key 6
  • F. Antonacci and P. Magrone, Second order nonautonomous systems with symmetric potential changing sign , Rend. Mat. Appl., 18(1998), 367–379 \ref\key 7
  • M. Badiale and E. Nabana, A remark on multiplicity of solutions for semilinear elliptic problems with indefinite nonlinearity , C. R. Acad. Sci. Paris Sér. I Math., 323(1996), 151–15 \ref\key 8
  • A. K. Ben–Naoum, C. Troestler and M. Willem, Existence and multiplicity results for homogeneous second order differential equations , J. Differential Equations, 112(1994), 239–249 \ref\key 9
  • H. Berestycki, I. Capuzzo–Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4(1994), 59–78 \ref\key 10 ––––, Variational methods for indefinite superlinear homogeneous elliptic problems , NoDEA Nonlinear Differential Equations Appl., 2(1995), 553–572 \ref\key 11
  • T. Burton and R. Grimmer, On continuability of solutions of second order differential equations , Proc. Amer. Math. Soc., 29(1971), 277–283 \ref\key 12
  • G. J. Butler, Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear differential equations , J. Differential Equations, 22(1976), 467–477 \ref\key 13 ––––, The existence of continuable solutions of a second order differential equation , Canad. J. Math., 29(1977), 472–479 \ref\key 14
  • P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1(1994), 97–129 \ref\key 15
  • A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics , in preparation \ref\key 16
  • A. Capietto, M. Henrard, J. Mawhin and F. Zanolin, A continuation approach to some forced superlinear Sturm–Liouville boundary value problems, Topol. Methods Nonlinear Anal., 3(1994), 81–100 \ref\key 17 ––––, A continuation approach to superlinear boundary value problems , J. Differential Equations, 88(1990), 347–395 \ref\key 18 ––––, On the existence of two solutions with a prescribed number of zeros for a superlinear two-point boundary value problem , Topol. Methods Nonlinear Anal., 6 (1995), 175–188 \ref\key 19
  • C. V. Coffman and D. F. Ullrich, On the continuability of solutions of a certain nonlinear differential Equation , Monatsh. Math., 71(1967), 385–392 \ref\key 20
  • T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential , J. Differential Equations, 97(1992), 328–378 \ref\key 22
  • Y. Ding and M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign , Dynamic Systems Appl., 2(1993), 131–145 \ref\key 23
  • H. Ehrmann, Über die Existenz der Lösungen von Randwertaufgaben bei gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung , (German, Math. Ann., 134 (1957), 167–194) \ref\key 24 ––––, Nachweis periodischer Lösungen bei gewissen nichtlinearen Schwingungsdifferentialgleichungen , (German, Arch. Rational Mech. Anal., 1 (1957), 124–138) \ref\key 25
  • G. Fei, The existence of homoclinic orbits for Hamiltonian systems with the potentials changing sign , Chinese Ann. Math. Ser. B, 17(1996), 403–410 ;, Chinese Ann. Math. Ser. A, 17(1996), 651–?, to appear, (Chinese summary) \ref\key 26
  • M. Girardi and M. Matzeu, Existence and multiplicity results for periodic solutions of superquadratic Hamiltonian systems where the potential changes sign, , NoDEA Nonlinear Differential Equations Appl., 2(1995), 35–61 \ref\key 27 ––––, Some results about periodic solutions of second order Hamiltonian systems where the potential has indefinite sign , Nonlinear Partial Differential Equations (Fès, 1994), Pitman Res. Notes Math. Ser., 343 , Longman, Harlow (1996), 147–154 \ref\key 28
  • P. Hartman, On boundary value problems for superlinear second order differential equations, , J. Differential Equations, 26(1977), 37–53 \ref\key 29
  • S. Khanfir and L. Lassoued, On the existence of positive solutions of a semilinear elliptic equation with change of sign, Nonlinear Anal., 22(1994), 1309–1314 \ref\key 30
  • S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergodic Theory Dynamic Systems, 11 (1991), 365–378 \ref\key 31
  • L. Lassoued, Periodic solutions of a second order superquadratic system with a change of sign in the potential , J. Differential Equations, 93(1991), 1–18 \ref\key 32
  • V. K. Le and K. Schmitt, Minimization problems for noncoercive functionals subject to constraints , Trans. Amer. Math. Soc., 347(1995), 4485–4513 \ref\key 33
  • M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials , Comm. Math. Phys., 143(1991), 43–83 \ref\key 34
  • M. Levi and E. Zehnder, Boundedness of solutions for quasiperiodic potentials , SIAM J. Math. Anal., 26(1995), 1233–1256 \ref\key 35
  • B. Liu, The stability of the equilibrium of a conservative system, J. Math. Anal. Appl., 202(1996), 133–149 \ref\key 36
  • J. Mawhin, Oscillatory properties of solutions and nonlinear differential equations with periodic boundary conditions , Second Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1992), Rocky Mountain J. Math., 25(1995), 7–37 \ref\key 37
  • G. R. Morris, A differential equation for undamped forced non-linear oscillations I, Proc. Cambridge Philos. Soc., 51(1955), 297–312 \ref\key 38 ––––, A differential equation for undamped forced non-linear oscillations II , Proc. Cambridge Philos. Soc., 54(1958), 426–438 \ref\key 39
  • Z. Nehari, Characteristic values associated with a class of nonlinear second order differential equations , Acta Math., 105(1961), 141–175 \ref\key 40
  • Z. Opial, Sur les périodes des solutions de l'équation différentielle $x\sp{\prime\prime}+g(x)=0$ , Ann. Polon. Math., 10(1961), 49–72 \ref\key 41
  • D. Papini, Infinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign , J. Math. Anal. Appl., 247 (2000), 217–235 \ref\key 42
  • M. Ramos, S. Terracini and C. Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities , J. Funct. Anal., 159(1998), 596–628 \ref\key 43
  • C. Rebelo and F. Zanolin, On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm–Liouville problems via the shooting map , Differential Integral Equations, to appear \ref\key 44
  • R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Math., 1458, Springer-Verlag, Berlin (1990) \ref\key 45
  • M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order , J. Differential Equation, 37(1980), 285–295 \ref\key 46
  • S. Terracini and G. Verzini, Oscillating solutions to second order ODE's with indefinite superlinear nonlinearities , Nonlinearity (1999), to appear \ref\key 47
  • P. Waltman, An oscillation criterion for a nonlinear second order equation , J. Math. Anal. Appl., 10(1965), 439–441