On a Concave–Convex Elliptic Problem with a Nonlinear Boundary Condition

We investigate an indefinite superlinear elliptic equation coupled with a sublinear Neumann boundary condition (depending on a positive parameter λ), which provides a concave–convex nature to the problem. We establish a global multiplicity result for positive solutions in the spirit of Ambrosetti–Brezis–Cerami and obtain their asymptotic profiles as λ→ 0. Furthermore, we also analyse the case where the nonlinearity is concave. Our arguments are based on a bifurcation analysis, a comparison principle, and variational techniques. © 2015, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.