  We obtain a pair of nontrivial solutions for a class of concave-linear-convex
type elliptic problems that are either critical or subcritical. The solutions
we find are neither local minimizers nor of mountain pass type in general. They
are higher critical points in the sense that they each have a higher critical
group that is nontrivial. This fact is crucial for showing that our solutions
are nontrivial. We also prove some intermediate results of independent interest
on the localization and homotopy invariance of critical groups of functionals
involving critical Sobolev exponents.
