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2015, 2015(special): 562-568. doi: 10.3934/proc.2015.0562

The Nehari solutions and asymmetric minimizers

Armands Gritsans 1, and  Felix Sadyrbaev 2,

1. 

Daugavpils University, Parades str. 1, Daugavpils, LV 5400, Latvia
2. 

Insitute of Mathematics and Computer Science, University of Latvia, Rainis boul. 29, Riga, LV 1459, Latvia

Received  September 2014 Revised  November 2014 Published  November 2015

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.
Keywords: Superlinear boundary value problem, asymmetric solutions., Nehari minimizer, bifurcation.
Mathematics Subject Classification: Primary: 34B15; Secondary: 34B1.
Citation: Armands Gritsans, Felix Sadyrbaev. The Nehari solutions and asymmetric minimizers. Conference Publications, 2015, 2015 (special) : 562-568. doi: 10.3934/proc.2015.0562
References:
[1]

Z.Nehari, Characteristic values associated with a class of nonlinear second order differential equations, Acta Math., 105 (1961), 141-176. MR0123775  Google Scholar

[2]

A. Gritsans and F. Sadyrbaev, Characteristic numbers of non-autonomous Emden-Fowler type equations, Mathematical Modelling and Analysis., 11 (2006), 243-252. MR2268126  Google Scholar

[3]

A. Gritsans and F. Sadyrbaev, Lemniscatic functions in the theory of the Emden - Fowler diferential equation, Mathematics. Differential equations (Univ. of Latvia, Institute of Math. and Comp. Sci.), 3: 5 - 27, 2003. (electr. version http://www.lumii.lv/Pages/sbornik/s3f3v1.pdf ). Google Scholar

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R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), no. 4, 1353-1362. MR2869119  Google Scholar

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F. Zh. Sadyrbaev, Solutions of an equation of Emden-Fowler type. (Russian), Differentsial'nye Uravneniya, 25 (1989), no. 5, 799-805; translation in Differential Equations 25 (1989), no. 5, 560-565. MR1003036  Google Scholar

show all references

References:
[1]

Z.Nehari, Characteristic values associated with a class of nonlinear second order differential equations, Acta Math., 105 (1961), 141-176. MR0123775  Google Scholar

[2]

A. Gritsans and F. Sadyrbaev, Characteristic numbers of non-autonomous Emden-Fowler type equations, Mathematical Modelling and Analysis., 11 (2006), 243-252. MR2268126  Google Scholar

[3]

A. Gritsans and F. Sadyrbaev, Lemniscatic functions in the theory of the Emden - Fowler diferential equation, Mathematics. Differential equations (Univ. of Latvia, Institute of Math. and Comp. Sci.), 3: 5 - 27, 2003. (electr. version http://www.lumii.lv/Pages/sbornik/s3f3v1.pdf ). Google Scholar

[4]

R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), no. 4, 1353-1362. MR2869119  Google Scholar

[5]

F. Zh. Sadyrbaev, Solutions of an equation of Emden-Fowler type. (Russian), Differentsial'nye Uravneniya, 25 (1989), no. 5, 799-805; translation in Differential Equations 25 (1989), no. 5, 560-565. MR1003036  Google Scholar

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