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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 & 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.  Google Scholar [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.  Google Scholar [3] E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192. doi: 10.1007/BF00282326.  Google Scholar [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.  Google Scholar [5] E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 173-179.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] E. N. Dancer, Real analyticity and non-degeneracy, Math. Ann., 325 (2003), 369-392.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1981.  Google Scholar [13] G. Whyburn, "Topological Analysis," Princeton University Press, Princeton, N. J., 1958. Google Scholar

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##### References:
 [1] K. C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems," Birkhäuser Boston Inc., Boston, 1993.  Google Scholar [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.  Google Scholar [3] E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181-192. doi: 10.1007/BF00282326.  Google Scholar [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.  Google Scholar [5] E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 173-179.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] E. N. Dancer, Real analyticity and non-degeneracy, Math. Ann., 325 (2003), 369-392.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1981.  Google Scholar [13] G. Whyburn, "Topological Analysis," Princeton University Press, Princeton, N. J., 1958. Google Scholar
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