# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 213-229. [2] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156. doi: 10.1016/0022-0396(88)90021-6. [3] E. N. Dancer, Domain variation for certain sets of solutions and applications, Topol. Methods Nonlinear Anal., 7 (1996), 95-113. [4] E. N. Dancer, On connecting orbits for competing species equations with large interactions, Topol. Methods Nonlinear Anal. 24 (2004), 1-19. [5] E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc. (3), 27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747. [6] E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations, 138 (1997), 86-132. doi: 10.1006/jdeq.1997.3256. [7] D. Daners, Domain perturbation for linear and semi-linear boundary value problems, in "Handbook of Differential Equations: Stationary Partial Differential Equations," Vol. {VI}, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 1-81. [8] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. [9] D. Henry, "Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations," With editorial assistance from Jack Hale and Antônio Luiz Pereira, London Mathematical Society Lecture Note Series, 318, Cambridge University Press, Cambridge, 2005. [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), 643-655. doi: 10.3934/dcds.2010.27.643. [11] K. P. Rybakowski, "The Homotopy Index and Partial Differential Equations," Universitext, Springer-Verlag, Berlin, 1987. [12] J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319.

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), 213-229. [2] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156. doi: 10.1016/0022-0396(88)90021-6. [3] E. N. Dancer, Domain variation for certain sets of solutions and applications, Topol. Methods Nonlinear Anal., 7 (1996), 95-113. [4] E. N. Dancer, On connecting orbits for competing species equations with large interactions, Topol. Methods Nonlinear Anal. 24 (2004), 1-19. [5] E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc. (3), 27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747. [6] E. N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations, 138 (1997), 86-132. doi: 10.1006/jdeq.1997.3256. [7] D. Daners, Domain perturbation for linear and semi-linear boundary value problems, in "Handbook of Differential Equations: Stationary Partial Differential Equations," Vol. {VI}, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 1-81. [8] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. [9] D. Henry, "Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations," With editorial assistance from Jack Hale and Antônio Luiz Pereira, London Mathematical Society Lecture Note Series, 318, Cambridge University Press, Cambridge, 2005. [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), 643-655. doi: 10.3934/dcds.2010.27.643. [11] K. P. Rybakowski, "The Homotopy Index and Partial Differential Equations," Universitext, Springer-Verlag, Berlin, 1987. [12] J.-C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319.
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