Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions

Abstract

Abstract This article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.