The Nehari solutions and asymmetric minimizers

Armands  Gritsans; Felix  Sadyrbaev

doi:10.3934/proc.2015.0562

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2015, Volume 2015: 562-568. Doi: 10.3934/proc.2015.0562 open access

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The Nehari solutions and asymmetric minimizers

Armands Gritsans^1, and
Felix Sadyrbaev^2,

1.
Daugavpils University, Parades str. 1, Daugavpils, LV 5400
2.
Insitute of Mathematics and Computer Science, University of Latvia, Rainis boul. 29, Riga, LV 1459

Received: September 2014

Revised: November 2014

Published: October 2015

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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.

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Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18.

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References

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