March  2011, 10(2): 785-802. doi: 10.3934/cpaa.2011.10.785

A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence

1. 

Quinnipiac University, 275 Mt Carmel Avenue, Hamden, CT 06518, United States

2. 

Department of Mathematics, 196 Auditorium Road, Unit 3009, University of Connecticut, Storrs, CT 06269-3009, United States

Received  January 2010 Revised  July 2010 Published  December 2010

We investigate a conjecture regarding the number of solutions of a second order elliptic boundary value problem with an asymmetric nonlinearity. This investigation makes use of several computer assisted techniques. First, we compute approximate solutions using Newton's Iteration for small $b$ and then use a continuation method to show that the number of solutions becomes large as $b$ increases.
Citation: Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure & Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785
References:
[1]

B. Breuer, P. J. McKenna and M. Plum, Multiple Solutions for a semilinear boundary value problem: a computational multiplicity proof,, Journal of Differential Equations, 195 (2003), 243. doi: doi:10.1016/S0022-0396(03)00186-4. Google Scholar

[2]

E. N. Dancer, A counterexample to the Lazer-McKenna conjecture,, Nonlinear Analysis, 13 (1989), 19. Google Scholar

[3]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture,, Journal of Differential Equations, 210 (2005), 317. doi: doi:10.1016/j.jde.2004.07.017. Google Scholar

[4]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture: Part II,, Communications in Partial Differential Equations, 30 (2005), 1331. doi: doi:10.1080/03605300500258865. Google Scholar

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E. N. Dancer and Sanjiban Santra, On the superlinear Lazer-McKenna conjecture: the non-homogeneous case,, Adv. Differential Equations, 12 (2007), 961. Google Scholar

[6]

Manuel del Pino and Claudio Muñoz, The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity,, J. Differential Equations, 231 (2006), 108. Google Scholar

[7]

O. Druet, The critical Lazer-McKenna conjecture in low dimensions,, Journal of Differential Equations, 245 (2008), 2199. doi: doi:10.1016/j.jde.2008.05.002. Google Scholar

[8]

Helmut Hofer, Variational and topological methods in partially ordered Hilbert Space,, Mathematishe Annalen, 261 (1982), 293. doi: doi:10.1007/BF01457453. Google Scholar

[9]

Gongbao Li, Shusen Yan and Jianfu Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth,, Calc. Var. Partial Differential Equations, 28 (2007), 471. doi: doi:10.1007/s00526-006-0051-z. Google Scholar

[10]

Riccardo Molle and Donato Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities,, Poincar\'e Anal. Non Lin\'eaire, 27 (2010), 529. Google Scholar

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Filomena Pacella and P. N. Srikanth, Nonradial solutions of a nonhomogeneous semilinear elliptic problem with linear growth,, J. Math. Anal. Appl., 341 (2008), 131. doi: doi:10.1016/j.jmaa.2007.09.059. Google Scholar

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J. Wei and S. Yan, Lazer-McKenna conjecture: the critical case, Journal of Functional Analysis, 244 (2007), 639. doi: doi:10.1016/j.jfa.2006.11.002. Google Scholar

show all references

References:
[1]

B. Breuer, P. J. McKenna and M. Plum, Multiple Solutions for a semilinear boundary value problem: a computational multiplicity proof,, Journal of Differential Equations, 195 (2003), 243. doi: doi:10.1016/S0022-0396(03)00186-4. Google Scholar

[2]

E. N. Dancer, A counterexample to the Lazer-McKenna conjecture,, Nonlinear Analysis, 13 (1989), 19. Google Scholar

[3]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture,, Journal of Differential Equations, 210 (2005), 317. doi: doi:10.1016/j.jde.2004.07.017. Google Scholar

[4]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture: Part II,, Communications in Partial Differential Equations, 30 (2005), 1331. doi: doi:10.1080/03605300500258865. Google Scholar

[5]

E. N. Dancer and Sanjiban Santra, On the superlinear Lazer-McKenna conjecture: the non-homogeneous case,, Adv. Differential Equations, 12 (2007), 961. Google Scholar

[6]

Manuel del Pino and Claudio Muñoz, The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity,, J. Differential Equations, 231 (2006), 108. Google Scholar

[7]

O. Druet, The critical Lazer-McKenna conjecture in low dimensions,, Journal of Differential Equations, 245 (2008), 2199. doi: doi:10.1016/j.jde.2008.05.002. Google Scholar

[8]

Helmut Hofer, Variational and topological methods in partially ordered Hilbert Space,, Mathematishe Annalen, 261 (1982), 293. doi: doi:10.1007/BF01457453. Google Scholar

[9]

Gongbao Li, Shusen Yan and Jianfu Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth,, Calc. Var. Partial Differential Equations, 28 (2007), 471. doi: doi:10.1007/s00526-006-0051-z. Google Scholar

[10]

Riccardo Molle and Donato Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities,, Poincar\'e Anal. Non Lin\'eaire, 27 (2010), 529. Google Scholar

[11]

Filomena Pacella and P. N. Srikanth, Nonradial solutions of a nonhomogeneous semilinear elliptic problem with linear growth,, J. Math. Anal. Appl., 341 (2008), 131. doi: doi:10.1016/j.jmaa.2007.09.059. Google Scholar

[12]

J. Wei and S. Yan, Lazer-McKenna conjecture: the critical case, Journal of Functional Analysis, 244 (2007), 639. doi: doi:10.1016/j.jfa.2006.11.002. Google Scholar

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