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**The Nehari solutions and asymmetric minimizers.**
*(English)*
Zbl 1381.34044

Authors’ abstract: We consider the boundary value problem \(x'' = -q(t,h) x^3,\) \(x(-1)=x(1)=0\) which exhibits bifurcation of the Nehari solutions. The Nehari solution of the problem is a solution which minimizes certain functional. We show that for \(h\) small there is exactly one Nehari solution. Then under the increase of \(h\) there appear two Nehari solutions which supply the functional smaller value than the remaining symmetrical solution does. So the bifurcation of the Nehari solutions is observed and the previously studied in the literature phenomenon of asymmetrical Nehari solutions is confirmed.

Reviewer: Haiyan Wang (Phoenix)

### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

49K05 | Optimality conditions for free problems in one independent variable |

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\textit{A. Gritsans} and \textit{F. Sadyrbaev}, Discrete Contin. Dyn. Syst. 2015, 562--568 (2015; Zbl 1381.34044)

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### References:

[1] | Z. Nehari, Characteristic values associated with a class of nonlinear second order differential equations,, Acta Math., 105, 141 (1961) · Zbl 0099.29104 |

[2] | A. Gritsans, Characteristic numbers of non-autonomous Emden-Fowler type equations,, Mathematical Modelling and Analysis., 11, 243 (2006) · Zbl 1107.34024 |

[3] | A. Gritsans, <em>Lemniscatic functions in the theory of the Emden - Fowler diferential equation</em>,, Mathematics. Differential equations (Univ. of Latvia (2003) |

[4] | R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation,, Proc. Amer. Math. Soc., 140, 1353 (2012) · Zbl 1248.34023 |

[5] | F. Zh. Sadyrbaev, Solutions of an equation of Emden-Fowler type. (Russian),, Differentsial’nye Uravneniya, 25, 799 (1989) · Zbl 0817.34011 |

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