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

Pages: 95 - 104, Issue special, November 2013

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Santiago Cano-Casanova - Dpto. de Matemática Aplicada y Computación, Escuela Técnica Superior de Ingeniería - ICAI, Universidad Pontificia Comillas, Alberto Aguilera, 25, 28015-Madrid, Spain (email)

Abstract: 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:  Nonlinear boundary conditions, bifurcation, positive solutions, nonlinear elliptic boundary value problems, logistic problems, mixed boundary conditions.
Mathematics Subject Classification:  Primary: 35J66, 35J65, 35J60; Secondary: 35J25.

Received: September 2012;      Revised: June 2013;      Published: November 2013.