  We consider a parametric nonlinear nonhomogeneous elliptic equation, driven
by the sum of two differential operators having different structure. The
associated energy functional has unbalanced growth and we do not impose any
global growth conditions to the reaction term, whose behavior is prescribed
only near the origin. Using truncation and comparison techniques and Morse
theory, we show that the problem has multiple solutions in the case of high
perturbations. We also show that if a symmetry condition is imposed to the
reaction term, then we can generate a sequence of distinct nodal solutions with
smaller and smaller energies.
