• Previous Article
    Continua of local minimizers in a quasilinear model of phase transitions
  • DCDS Home
  • This Issue
  • Next Article
    On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian
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 - A, 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., (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), 3.  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.  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.   Google Scholar

[5]

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

[6]

E. N. Dancer, Finite Morse index solutions of supercritical problems,, J. Reine Angew. Math., 620 (2008), 213.  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.  doi: 10.1016/0022-0396(80)90107-2.  Google Scholar

[8]

E. N. Dancer, Real analyticity and non-degeneracy,, Math. Ann., 325 (2003), 369.   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.  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.  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.  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, (1981).   Google Scholar

[13]

G. Whyburn, "Topological Analysis,", Princeton University Press, (1958).   Google Scholar

show all references

References:
[1]

K. C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems,", Birkhäuser Boston Inc., (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), 3.  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.  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.   Google Scholar

[5]

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

[6]

E. N. Dancer, Finite Morse index solutions of supercritical problems,, J. Reine Angew. Math., 620 (2008), 213.  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.  doi: 10.1016/0022-0396(80)90107-2.  Google Scholar

[8]

E. N. Dancer, Real analyticity and non-degeneracy,, Math. Ann., 325 (2003), 369.   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.  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.  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.  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, (1981).   Google Scholar

[13]

G. Whyburn, "Topological Analysis,", Princeton University Press, (1958).   Google Scholar

[1]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[2]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[3]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231

[4]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439

[5]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

[6]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[7]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[8]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[9]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[10]

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046

[11]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[12]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318

[13]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[14]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229

[15]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053

[16]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[17]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[18]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010

[19]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[20]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]