Nonlinear mixed boundary conditions in BVPs of Logistic type with spatial heterogeneities and a nonlinear flux on the boundary with arbitrary sign. The case p>2q−1p>2q−1

This paper analyzes the structure of the set of positive solutions of a general class of BVPs of Logistic type, with spatial heterogeneities and nonlinear mixed boundary conditions containing a flux with arbitrary sign. The main results obtained depend on the nodal behavior of the potential appearing in the partial differential equation, on the sign of the nonlinear flux on the boundary and on the relationship the exponents p>1p>1, from the reaction in the partial differential equation, and q>1q>1, from the nonlinear flux on the boundary. This paper is focused to the case p>2q−1p>2q−1. The main technical tools used are the Characterization of the Strong Maximum Principle given by H. Amann and J. López-Gómez [3], monotonicity technics, blow-up arguments and bifurcation. The results obtained in this paper are the natural extensions or complements of the previous ones obtained by S. Cano-Casanova [7] and [8], C. Morales-Rodrigo and A. Suárez [17] and J. García-Melián, C. Morales-Rodrigo, J.D. Rossi and A. Suárez [11].