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Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight. (English) Zbl 1218.34027
Summary: We are concerned with the global structure and stability of positive solutions to the periodic boundary value problem $$-u''(t)+q(t)u(t)=\lambda a(t)f(u(t)),\quad 0<t<2\pi,\quad u(0)=u(2\pi),\quad u'(0)=u'(2\pi),$$ where $q\in C(\Bbb R,[0,\infty))$ is of period $2\pi$ and $q(t)\equiv 0$, $t\in[0,2\pi]$; $a\in C(\Bbb R,\Bbb R)$ is of period $2\pi$ and changes sign. The proof of our main results are based on bifurcation techniques.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34C23Bifurcation (ODE)
34C25Periodic solutions of ODE
Full Text: DOI
[1] Camassa, R.; Holm, D.: An integrable shallow water equation with peaked solitons, Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[2] Alber, M. S.; Camassa, R.; Holm, D.; Marsden, J. E.: The geometry of peaked solitons of a class of integrable PDE’s, Lett. math. Phys. 32, 37-151 (1994) · Zbl 0808.35124 · doi:10.1007/BF00739423
[3] Camassa, R.; Holm, D.; Hyman, J.: A new integrable shallow water equation, Adv. appl. Mech. 31, 1-33 (1994) · Zbl 0808.76011
[4] Constantin, A.; Mckean, H. P.: A shallow water equation on the circle, Commun. pure appl. Math. 52, 949-982 (1999) · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[5] Constantin, A.: On the spectral problem for the periodic Camassa--Holm equation, J. math. Anal. appl. 210, 215-230 (1997) · Zbl 0881.35102 · doi:10.1006/jmaa.1997.5393
[6] Johnson, R. S.: Camassa--Holm, Korteweg--de Vries and related models for water waves, J. fluid mech. 455, 63-82 (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[7] Constantin, A.; Lannes, D.: The hydrodynamical relevance of the Camassa--Holm and Degasperis--Procesi equations, Arch. ration. Mech. anal. 192, 165-186 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[8] Lakshmanan, M.: Integrable nonlinear wave equations and possible connections to tsunami dynamics, in tsunami and nonlinear waves, (2007) · Zbl 1310.76044
[9] Constantin, A.; Johnson, R. S.: Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid dynam. Res. 40, 175-211 (2008) · Zbl 1135.76007 · doi:10.1016/j.fluiddyn.2007.06.004
[10] Atici, F. M.; Guseinov, G. Sh.: On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. comput. Appl. math. 132, 341-356 (2001) · Zbl 0993.34022 · doi:10.1016/S0377-0427(00)00438-6
[11] Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations 190, No. 2, 643-662 (2003) · Zbl 1032.34040 · doi:10.1016/S0022-0396(02)00152-3
[12] Graef, J. R.; Kong, L.; Wang, H.: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. differential equations 245, No. 5, 1185-1197 (2008) · Zbl 1203.34028 · doi:10.1016/j.jde.2008.06.012
[13] Hao, X. N.; Liu, L. S.; Wu, Y. H.: Existence and multiplicity results for nonlinear periodic boundary value problems, Nonlinear anal. 72, 3635-3642 (2010) · Zbl 1195.34033 · doi:10.1016/j.na.2009.12.044
[14] Kielhöfer, H.: Bifurcation theory: an introduction with applications to pdes, (2004) · Zbl 1032.35001
[15] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[16] Constantin, Adrian: A general-weighted Sturm--Liouville problem, Estratto dagli annali Della scuola normale superiore di Pisa scienze fisiche e matematiche e matematiche -- serie IV. Vol. XXIV. Fasc. 4, 767-782 (1997) · Zbl 0913.34022 · numdam:ASNSP_1997_4_24_4_767_0
[17] Ma, R. Y.; Xu, J.: Bifurcation from interval and positive solutions for second order periodic boundary value problems, Dyn. syst. Appl. 19, 211-224 (2010) · Zbl 1218.34026
[18] Ma, R. Y.; Xu, J.: Bifurcation from interval and positive solutions for second order periodic boundary value problems, Appl. math. Comput. 216, No. 18, 2463-2471 (2010) · Zbl 1210.34035 · doi:10.1016/j.amc.2010.03.092
[19] Rabinowitz, P. H.: On bifurcation from infinity, J. differential equations 14, 462-475 (1973) · Zbl 0272.35017 · doi:10.1016/0022-0396(73)90061-2