Equations with unbalanced growth and logistic type reaction

We consider a Dirichlet problem driven by a nonautonomous (p, q)differential operator with unbalanced growth. The reaction is of logistic-type, but in contrast to the standard logistic equation, both parts of the reaction are strictly "sublinear" (concave) with respect to the exponents of the differential operator. Using critical point theory, truncation and comparison techniques, and critical groups, we prove an existence and multiplicity theorem which is global in the parameter lambda > 0 (a bifurcation-type theorem).