Multiplicity of solutions for a convex-concave problem with a nonlinear boundary condition

We study the existence of multiple positive solutions for a convex-concave problem, denoted by (P-lambda mu), with a nonlinear boundary condition involving two critical exponents and two positive parameters lambda and mu. We obtain a continuous strictly decreasing function f such that K-1 = {(f(mu), mu) : mu is an element of [0, infinity)} divides [0, infinity) x [0, infinity) \ {(0, 0)} in two connected sets K-0 and K-2 such that problem (P-lambda mu) has at least two solutions for (lambda,mu) is an element of K-2, at least one solution for (lambda, mu) is an element of K-1 and no solution for (lambda,mu) is an element of K-0.