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On the global regularity of axisymmetric Navier-Stokes-Boussinesq system
Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator
1. | Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires, Argentina, Argentina |
$u'' + m^2 u + g(u)=p(t),\qquad u(0)-u(2\pi)=u'(0)-u'(2\pi)=0$
admits at least one solution, provided that
$(\a_m(p)^2+$β$_m(p)^2$$)^\frac 1\2$< $\frac 2\pi |g(+\infty)-g(-\infty)|,$
where $\a_m(p)$ and β$_m(p)$ denote the $m$-th Fourier coefficients of
the forcing
term $p$.
In this article we prove that, as it occurs in the case $m=0$,
the condition on $g$ may be relaxed. In particular,
no specific behavior at infinity is assumed.
References:
[1] |
P. Amster and P. De Nápoli, On a generalization of Lazer-Leach conditions for a system of second order ODE's, Topological Methods in Nonlinear Analysis, 33 (2009), 31-39. |
[2] |
D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA., 28 (1997), 1623-1632.
doi: 10.1016/S0362-546X(96)00022-3. |
[3] |
C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78.
doi: 10.1006/jdeq.1998.3441. |
[4] |
C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. (4), 157 (1990), 99-116.
doi: 10.1007/BF01765314. |
[5] |
C. Fabry and C. Franchetti, Nonlinear equations with growth restrictions on the nonlinear term, J. Differential Equations, 20 (1976), 283-291.
doi: 10.1016/0022-0396(76)90108-X. |
[6] |
C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity, 13 (2000), 493-505.
doi: 10.1088/0951-7715/13/3/302. |
[7] |
A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Mathematical and Computer Modelling, 32 (2000), 1445-1455.
doi: 10.1016/S0895-7177(00)00216-8. |
[8] |
E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623. |
[9] |
A. Lazer, On Schauder's fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl.,21 (1968), 421-425.
doi: 10.1016/0022-247X(68)90225-4. |
[10] |
A. Lazer and D. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68.
doi: 10.1007/BF02410787. |
[11] |
J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems," NSF-CBMS Regional Conference in Mathematics 40, American Mathematical Society, Providence, RI, 1979. |
[12] |
J. Mawhin, Landesman-Lazer conditions for boundary value problems: A nonlinear version of resonance, Bol. de la Sociedad Española de Mat. Aplicada, 16 (2000), 45-65. |
[13] |
L. Nirenberg, Generalized degree and nonlinear problems, in "Contributions to Nonlinear Functional Analysis" (ed. E. H. Zarantonello), Academic Press, New York, (1971), 1-9. |
[14] |
R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc., 34 (2002), 308-318.
doi: 10.1112/S0024609301008748. |
[15] |
R. Ortega and J. R. Ward Jr., A semilinear elliptic system with vanishing nonlinearities, Discrete and Continuous Dynamical Systems: A Supplement Volume "Dynamical Systems and Differential Equations," (2003), 688-693. |
[16] |
D. Ruiz and J. R. Ward Jr., Some notes on periodic systems with linear part at resonance, Discrete and Continuous Dynamical Systems, 11 (2004), 337-350.
doi: 10.3934/dcds.2004.11.337. |
show all references
References:
[1] |
P. Amster and P. De Nápoli, On a generalization of Lazer-Leach conditions for a system of second order ODE's, Topological Methods in Nonlinear Analysis, 33 (2009), 31-39. |
[2] |
D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA., 28 (1997), 1623-1632.
doi: 10.1016/S0362-546X(96)00022-3. |
[3] |
C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78.
doi: 10.1006/jdeq.1998.3441. |
[4] |
C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. (4), 157 (1990), 99-116.
doi: 10.1007/BF01765314. |
[5] |
C. Fabry and C. Franchetti, Nonlinear equations with growth restrictions on the nonlinear term, J. Differential Equations, 20 (1976), 283-291.
doi: 10.1016/0022-0396(76)90108-X. |
[6] |
C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity, 13 (2000), 493-505.
doi: 10.1088/0951-7715/13/3/302. |
[7] |
A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Mathematical and Computer Modelling, 32 (2000), 1445-1455.
doi: 10.1016/S0895-7177(00)00216-8. |
[8] |
E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623. |
[9] |
A. Lazer, On Schauder's fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl.,21 (1968), 421-425.
doi: 10.1016/0022-247X(68)90225-4. |
[10] |
A. Lazer and D. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68.
doi: 10.1007/BF02410787. |
[11] |
J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems," NSF-CBMS Regional Conference in Mathematics 40, American Mathematical Society, Providence, RI, 1979. |
[12] |
J. Mawhin, Landesman-Lazer conditions for boundary value problems: A nonlinear version of resonance, Bol. de la Sociedad Española de Mat. Aplicada, 16 (2000), 45-65. |
[13] |
L. Nirenberg, Generalized degree and nonlinear problems, in "Contributions to Nonlinear Functional Analysis" (ed. E. H. Zarantonello), Academic Press, New York, (1971), 1-9. |
[14] |
R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc., 34 (2002), 308-318.
doi: 10.1112/S0024609301008748. |
[15] |
R. Ortega and J. R. Ward Jr., A semilinear elliptic system with vanishing nonlinearities, Discrete and Continuous Dynamical Systems: A Supplement Volume "Dynamical Systems and Differential Equations," (2003), 688-693. |
[16] |
D. Ruiz and J. R. Ward Jr., Some notes on periodic systems with linear part at resonance, Discrete and Continuous Dynamical Systems, 11 (2004), 337-350.
doi: 10.3934/dcds.2004.11.337. |
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