THE COMPETITION BETWEEN INCOMING AND OUTGOING FLUXES IN AN ELLIPTIC PROBLEM

Abstract

In this work, we consider existence and uniqueness of positive solutions to the elliptic equation -Δu = λu in Ω, with the nonlinear boundary conditions [Formula: see text] on Γ1, [Formula: see text] on Γ2, where Ω is a smooth bounded domain, ∂Ω = Γ1 ∪ Γ2, [Formula: see text], ν is the outward unit normal, p, q > 0 and λ is a real parameter. We obtain a complete picture of the bifurcation diagram of the problem, depending on the values of p, q and the parameter λ. Our proofs are based on different techniques: variational arguments, bifurcation techniques or comparison arguments, depending on the range of parameters considered.