Linear elliptic and parabolic PDEs with nonlinear mixed boundary conditions and spatial heterogeneities

This article concerns the positive solutions of a boundary-value problem constituted by a linear elliptic partial differential equation, subject to nonlinear mixed boundary conditions containing spatial heterogeneities with arbitrary sign along the boundary. The results obtained in this work provide us the global bifurcation diagram of positive solutions, the pointing behavior of them when the parameters change and the dynamics of the positive solutions of the associated parabolic problem. The main contribution of this paper is to give general results about existence, uniqueness, stability and pointing behavior of positive solutions, for boundary-value problems with nonlinear boundary conditions of mixed type containing spatial heterogeneities. The main technical tools used to develop the mathematical analysis are local and global bifurcation, monotonicity techniques, the Characterization of the Strong Maximum Principle given by Amann and Lopez-Gomez [5] blow up arguments and some of the techniques used in the previous works [19,20,33,34]. The results obtained in this paper are the natural continuation of the previous ones in [11]