• Previous Article
    Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment
  • DCDS Home
  • This Issue
  • Next Article
    Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
February  2016, 36(2): 941-952. doi: 10.3934/dcds.2016.36.941

The effects of spatial heterogeneities on some multiplicity results

1. 

Department of Applied Mathematics, Complutense University of Madrid, Madrid, 28040

2. 

Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, United States

Received  June 2014 Revised  October 2014 Published  August 2015

In [10], using a Theorem of Clark and in [1] several multiplicity results were obtained for families of semilinear elliptic partial differential equations. Here these results are extended so as to include more general spatially heterogeneous models arising in population dynamics. The optimality of the general assumptions imposed to get some of these multiplicity results is also analyzed.
Citation: Julián López-Gómez, Paul H. Rabinowitz. The effects of spatial heterogeneities on some multiplicity results. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 941-952. doi: 10.3934/dcds.2016.36.941
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems,, Clarendon Press, (1998).   Google Scholar

[3]

D. C. Clark, A variant of the Ljusternik-Shnirelmann theory,, Indiana Univ. Math. J., 22 (1972), 65.  doi: 10.1512/iumj.1973.22.22008.  Google Scholar

[4]

D. de Figueiredo, Positive solutions of semilinear elliptic problems,, Lectures Notes in Mathematics, 957 (1982), 34.   Google Scholar

[5]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Diff. Eqns., 5 (1980), 999.  doi: 10.1080/03605308008820162.  Google Scholar

[6]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra,, in Handbook of Differential Equations, (2005), 211.  doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[7]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine,, Boll. Un. Ma. Ital., 7 (1973), 285.   Google Scholar

[8]

P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Comm. Pure Appl. Math., {23 (1970), 939.  doi: 10.1002/cpa.3160230606.  Google Scholar

[9]

P. H. Rabinowitz, A note on pairs of solutions of a nonlinear Sturm-Liouville problem,, Manuscripta Math., 11 (1974), 273.  doi: 10.1007/BF01173718.  Google Scholar

[10]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, Conference board of the mathematical sciences. Regional conference series in mathematics 65, (1986).   Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems,, Clarendon Press, (1998).   Google Scholar

[3]

D. C. Clark, A variant of the Ljusternik-Shnirelmann theory,, Indiana Univ. Math. J., 22 (1972), 65.  doi: 10.1512/iumj.1973.22.22008.  Google Scholar

[4]

D. de Figueiredo, Positive solutions of semilinear elliptic problems,, Lectures Notes in Mathematics, 957 (1982), 34.   Google Scholar

[5]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Diff. Eqns., 5 (1980), 999.  doi: 10.1080/03605308008820162.  Google Scholar

[6]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra,, in Handbook of Differential Equations, (2005), 211.  doi: 10.1016/S1874-5733(05)80012-9.  Google Scholar

[7]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine,, Boll. Un. Ma. Ital., 7 (1973), 285.   Google Scholar

[8]

P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Comm. Pure Appl. Math., {23 (1970), 939.  doi: 10.1002/cpa.3160230606.  Google Scholar

[9]

P. H. Rabinowitz, A note on pairs of solutions of a nonlinear Sturm-Liouville problem,, Manuscripta Math., 11 (1974), 273.  doi: 10.1007/BF01173718.  Google Scholar

[10]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, Conference board of the mathematical sciences. Regional conference series in mathematics 65, (1986).   Google Scholar

[1]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011

[2]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367

[3]

Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313

[4]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109

[5]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455

[6]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413

[7]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555

[8]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43

[9]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945

[10]

Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611

[11]

Koray Karabina, Edward Knapp, Alfred Menezes. Generalizations of Verheul's theorem to asymmetric pairings. Advances in Mathematics of Communications, 2013, 7 (1) : 103-111. doi: 10.3934/amc.2013.7.103

[12]

Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020

[13]

Shalosh B. Ekhad and Doron Zeilberger. Proof of Conway's lost cosmological theorem. Electronic Research Announcements, 1997, 3: 78-82.

[14]

Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719

[15]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067

[16]

Yves Coudière, Anđela Davidović, Clair Poignard. Modified bidomain model with passive periodic heterogeneities. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020126

[17]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990

[18]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[19]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619

[20]

Ben Green, Terence Tao, Tamar Ziegler. An inverse theorem for the Gowers $U^{s+1}[N]$-norm. Electronic Research Announcements, 2011, 18: 69-90. doi: 10.3934/era.2011.18.69

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]