Abstract
We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.
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Umezu, K. Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 17, 323–336 (2010). https://doi.org/10.1007/s00030-010-0056-3
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DOI: https://doi.org/10.1007/s00030-010-0056-3