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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 and Continuous Dynamical Systems, 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), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, Clarendon Press, Oxford, 1998.

[3]

D. C. Clark, A variant of the Ljusternik-Shnirelmann theory, Indiana Univ. Math. J., 22 (1972), 65-74. doi: 10.1512/iumj.1973.22.22008.

[4]

D. de Figueiredo, Positive solutions of semilinear elliptic problems, Lectures Notes in Mathematics, Springer, 957 (1982), 34-87.

[5]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Diff. Eqns., 5 (1980), 999-1030. doi: 10.1080/03605308008820162.

[6]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra, in Handbook of Differential Equations "Stationary Partial Differential Equations", edited by M. Chipot and P. Quittner, Elsevier Science B. V., North Holland, Chapter 4, Vol. II, pp. 211-309, Amsterdam 2005. doi: 10.1016/S1874-5733(05)80012-9.

[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), 285-301.

[8]

P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure Appl. Math., {23} (1970), 939-961. doi: 10.1002/cpa.3160230606.

[9]

P. H. Rabinowitz, A note on pairs of solutions of a nonlinear Sturm-Liouville problem, Manuscripta Math., 11 (1974), 273-282. doi: 10.1007/BF01173718.

[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, Amer. Math. Soc., Providence, RI, 1986.

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), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, Clarendon Press, Oxford, 1998.

[3]

D. C. Clark, A variant of the Ljusternik-Shnirelmann theory, Indiana Univ. Math. J., 22 (1972), 65-74. doi: 10.1512/iumj.1973.22.22008.

[4]

D. de Figueiredo, Positive solutions of semilinear elliptic problems, Lectures Notes in Mathematics, Springer, 957 (1982), 34-87.

[5]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Diff. Eqns., 5 (1980), 999-1030. doi: 10.1080/03605308008820162.

[6]

J. López-Gómez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra, in Handbook of Differential Equations "Stationary Partial Differential Equations", edited by M. Chipot and P. Quittner, Elsevier Science B. V., North Holland, Chapter 4, Vol. II, pp. 211-309, Amsterdam 2005. doi: 10.1016/S1874-5733(05)80012-9.

[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), 285-301.

[8]

P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure Appl. Math., {23} (1970), 939-961. doi: 10.1002/cpa.3160230606.

[9]

P. H. Rabinowitz, A note on pairs of solutions of a nonlinear Sturm-Liouville problem, Manuscripta Math., 11 (1974), 273-282. doi: 10.1007/BF01173718.

[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, Amer. Math. Soc., Providence, RI, 1986.

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