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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)=λa(t)f(u(t)),0<t<2π,u(0)=u(2π),u ' (0)=u ' (2π),

where qC(,[0,)) is of period 2π and q(t)0, t[0,2π]; aC(,) is of period 2π and changes sign. The proof of our main results are based on bifurcation techniques.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34C23Bifurcation (ODE)
34C25Periodic solutions of ODE
References:
[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)
[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)
[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 · doi: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