November  2012, 32(11): 3861-3869. doi: 10.3934/dcds.2012.32.3861

On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su

1. 

School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia

Received  February 2011 Revised  June 2011 Published  June 2012

We prove a new domain variation result for Neumann problems and apply it to give new examples of non-uniqueness of positive solutions.
Citation: E. N. Dancer. On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3861-3869. doi: 10.3934/dcds.2012.32.3861
References:
[1]

T. Bartsch and E. N. Dancer, Poincaré-Hopf type formulas on convex sets of Banach spaces,, Topol. Methods Nonlinear Anal., 34 (2009), 213. Google Scholar

[2]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, J. Differential Equations, 74 (1988), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar

[3]

E. N. Dancer, Domain variation for certain sets of solutions and applications,, Topol. Methods Nonlinear Anal., 7 (1996), 95. Google Scholar

[4]

E. N. Dancer, On connecting orbits for competing species equations with large interactions,, Topol. Methods Nonlinear Anal. \textbf{24} (2004), 24 (2004), 1. Google Scholar

[5]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems,, Proc. London Math. Soc. (3), 27 (1973), 747. doi: 10.1112/plms/s3-27.4.747. Google Scholar

[6]

E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions,, J. Differential Equations, 138 (1997), 86. doi: 10.1006/jdeq.1997.3256. Google Scholar

[7]

D. Daners, Domain perturbation for linear and semi-linear boundary value problems,, in, (2008), 1. Google Scholar

[8]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, 224 (1983). Google Scholar

[9]

D. Henry, "Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations,", With editorial assistance from Jack Hale and Ant\^onio Luiz Pereira, 318 (2005). Google Scholar

[10]

Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. {II. Stability and multiplicity},, Discrete Contin. Dyn. Syst., 27 (2010), 643. doi: 10.3934/dcds.2010.27.643. Google Scholar

[11]

K. P. Rybakowski, "The Homotopy Index and Partial Differential Equations,", Universitext, (1987). Google Scholar

[12]

J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems,, Comm. Partial Differential Equations, 4 (1979), 293. Google Scholar

show all references

References:
[1]

T. Bartsch and E. N. Dancer, Poincaré-Hopf type formulas on convex sets of Banach spaces,, Topol. Methods Nonlinear Anal., 34 (2009), 213. Google Scholar

[2]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, J. Differential Equations, 74 (1988), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar

[3]

E. N. Dancer, Domain variation for certain sets of solutions and applications,, Topol. Methods Nonlinear Anal., 7 (1996), 95. Google Scholar

[4]

E. N. Dancer, On connecting orbits for competing species equations with large interactions,, Topol. Methods Nonlinear Anal. \textbf{24} (2004), 24 (2004), 1. Google Scholar

[5]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems,, Proc. London Math. Soc. (3), 27 (1973), 747. doi: 10.1112/plms/s3-27.4.747. Google Scholar

[6]

E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions,, J. Differential Equations, 138 (1997), 86. doi: 10.1006/jdeq.1997.3256. Google Scholar

[7]

D. Daners, Domain perturbation for linear and semi-linear boundary value problems,, in, (2008), 1. Google Scholar

[8]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, 224 (1983). Google Scholar

[9]

D. Henry, "Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations,", With editorial assistance from Jack Hale and Ant\^onio Luiz Pereira, 318 (2005). Google Scholar

[10]

Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. {II. Stability and multiplicity},, Discrete Contin. Dyn. Syst., 27 (2010), 643. doi: 10.3934/dcds.2010.27.643. Google Scholar

[11]

K. P. Rybakowski, "The Homotopy Index and Partial Differential Equations,", Universitext, (1987). Google Scholar

[12]

J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems,, Comm. Partial Differential Equations, 4 (1979), 293. Google Scholar

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