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2013, 2013(special): 95-104. doi: 10.3934/proc.2013.2013.95

Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions

Santiago Cano-Casanova 1,

1. 

Dpto. de Matemática Aplicada y Computación, Escuela Técnica Superior de Ingeniería - ICAI, Universidad Pontificia Comillas, Alberto Aguilera, 25, 28015-Madrid, Spain

Received  September 2012 Revised  June 2013 Published  November 2013

In this paper a nonlinear boundary value problem of logistic type is considered, with nonlinear mixed boundary conditions, and with spatial heterogeneities of arbitrary sign in the differential equation and on the boundary conditions. The main goal of this paper is analyzing the structure of the continuum of positive solutions emanating from the trivial state at a unique bifurcation value, depending on the size and sign of the different potentials and parameters of the problem. The results in this paper extend the previous ones obtained by R. Gómez-Reñasco and J. López-Gómez [5, Proposition 2.1], for a superlinear indefinite problem of logistic type under Dirichlet boundary conditions, to a wide class of superlinear indefinite problems with nonlinear indefinite mixed boundary conditions.
Keywords: positive solutions, nonlinear elliptic boundary value problems, mixed boundary conditions., Nonlinear boundary conditions, bifurcation, logistic problems.
Mathematics Subject Classification: Primary: 35J66, 35J65, 35J60; Secondary: 35J2.
Citation: Santiago Cano-Casanova. Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions. Conference Publications, 2013, 2013 (special) : 95-104. doi: 10.3934/proc.2013.2013.95
References:
[1]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, in "New Developments in Differential Equations" (ed. W. Eckhaus), Math Studies, 21, North-Holland, Amsterdam (1976), 43-63.  Google Scholar

[2]

H. Amann, Dual semigroups second order linear elliptic boundary value problems, Israel Journal of Mathematics, 45 (1983), 225-254.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Analysis, 8 (1971), 321-340.  Google Scholar

[4]

J. García-Melián, C. Morales-Rodrigo, J. D. Rossi and A. Suárez, Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary, Ann. Mat. Pura Appl. (4) , 187 (2008), 459-486.  Google Scholar

[5]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffucion equations, Journal of Differential Equations , 167 (2000), 36-72.  Google Scholar

[6]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions", Princeton Mathematical Series, No. 30. Princeton Univ. Press, Princeton, NJ, 1970.  Google Scholar

show all references

References:
[1]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, in "New Developments in Differential Equations" (ed. W. Eckhaus), Math Studies, 21, North-Holland, Amsterdam (1976), 43-63.  Google Scholar

[2]

H. Amann, Dual semigroups second order linear elliptic boundary value problems, Israel Journal of Mathematics, 45 (1983), 225-254.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Analysis, 8 (1971), 321-340.  Google Scholar

[4]

J. García-Melián, C. Morales-Rodrigo, J. D. Rossi and A. Suárez, Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary, Ann. Mat. Pura Appl. (4) , 187 (2008), 459-486.  Google Scholar

[5]

R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffucion equations, Journal of Differential Equations , 167 (2000), 36-72.  Google Scholar

[6]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions", Princeton Mathematical Series, No. 30. Princeton Univ. Press, Princeton, NJ, 1970.  Google Scholar

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