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January  2013, 33(1): 153-161. doi: 10.3934/dcds.2013.33.153

Some bifurcation results for rapidly growing nonlinearities

1. 

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Received  August 2011 Revised  January 2012 Published  September 2012

We prove results on when nonlinear elliptic equations have infinitely many bifurcations if the nonlinearities grow rapidly.
Citation: E. N. Dancer. Some bifurcation results for rapidly growing nonlinearities. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 153-161. doi: 10.3934/dcds.2013.33.153
References:
[1]

K. C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems," Birkhäuser Boston Inc., Boston, 1993.

[2]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc., s3-27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747.

[3]

E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192. doi: 10.1007/BF00282326.

[4]

E. N. Dancer, Stable solutions on $\mathbb R^n$ and the primary branch of some non-self-adjoint convex problems, Differential Integral Equations, 17 (2004), 961-970.

[5]

E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 173-179.

[6]

E. N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math., 620 (2008), 213-233. doi: 10.1515/CRELLE.2008.055.

[7]

E. N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations, 37 (1980), 404-437. doi: 10.1016/0022-0396(80)90107-2.

[8]

E. N. Dancer, Real analyticity and non-degeneracy, Math. Ann., 325 (2003), 369-392.

[9]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbf R^n$ or on bounded domains with small diffusion, Trans. Amer. Math. Soc., 357 (2005), 1225-1243. doi: 10.1090/S0002-9947-04-03543-3.

[10]

E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4), 178 (2000), 225-233. doi: 10.1007/BF02505896.

[11]

E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u=e^u$ on $\mathbb R^N$: stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338. doi: 10.1090/S0002-9939-08-09772-4.

[12]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1981.

[13]

G. Whyburn, "Topological Analysis," Princeton University Press, Princeton, N. J., 1958.

show all references

References:
[1]

K. C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems," Birkhäuser Boston Inc., Boston, 1993.

[2]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc., s3-27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747.

[3]

E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192. doi: 10.1007/BF00282326.

[4]

E. N. Dancer, Stable solutions on $\mathbb R^n$ and the primary branch of some non-self-adjoint convex problems, Differential Integral Equations, 17 (2004), 961-970.

[5]

E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 173-179.

[6]

E. N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math., 620 (2008), 213-233. doi: 10.1515/CRELLE.2008.055.

[7]

E. N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations, 37 (1980), 404-437. doi: 10.1016/0022-0396(80)90107-2.

[8]

E. N. Dancer, Real analyticity and non-degeneracy, Math. Ann., 325 (2003), 369-392.

[9]

E. N. Dancer, Stable and finite Morse index solutions on $\mathbf R^n$ or on bounded domains with small diffusion, Trans. Amer. Math. Soc., 357 (2005), 1225-1243. doi: 10.1090/S0002-9947-04-03543-3.

[10]

E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4), 178 (2000), 225-233. doi: 10.1007/BF02505896.

[11]

E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u=e^u$ on $\mathbb R^N$: stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338. doi: 10.1090/S0002-9939-08-09772-4.

[12]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1981.

[13]

G. Whyburn, "Topological Analysis," Princeton University Press, Princeton, N. J., 1958.

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