  We consider a nonlinear Robin problem driven by a nonhomogeneous differential
operator, with reaction which exhibits the competition of two Carath\'eodory
terms. One is parametric, $(p-1)$-sublinear with a partially concave
nonlinearity near zero. The other is $(p-1)$-superlinear and has almost
critical growth. Exploiting the special geometry of the problem, we prove a
bifurcation-type result, describing the changes in the set of positive
solutions as the parameter $\lambda>0$ varies.
