American Institute of Mathematical Sciences

Advanced
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

[2]

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

[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 ).

[4]

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

[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

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

[2]

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

[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 ).

[4]

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

[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

[1]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
[2]

M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure and Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411
[3]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[4]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[5]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[6]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[7]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[8]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276
[9]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337
[10]

Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic and Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029
[11]

Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263
[12]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks and Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
[13]

John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291
[14]

John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89
[15]

Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
[16]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[17]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269
[18]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416
[19]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
[20]

Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103

 Impact Factor: 

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy