Some bifurcation results for rapidly growing nonlinearities

E. N. Dancer

doi:10.3934/dcds.2013.33.153

Article Contents

2013, Volume 33, Issue 1: 153-161. Doi: 10.3934/dcds.2013.33.153

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Some bifurcation results for rapidly growing nonlinearities

E. N. Dancer^1,

1.
School of Mathematics and Statistics, The University of Sydney, NSW 2006

Received: August 2011

Revised: January 2012

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

We prove results on when nonlinear elliptic equations have infinitely many bifurcations if the nonlinearities grow rapidly.

Keywords:

Nonlinear elliptic equations with rapidly growing nonlinearities,
many bifurcations.

Mathematics Subject Classification: Primary: 35A15, 35J25, 47H50, 58E99.

Citation:

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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.