  In this paper, by utilizing a newly established variational principle on
convex sets, we provide an existence and multiplicity result for a class of
semilinear elliptic problems defined on the whole $\mathbb R^N$ with
nonlinearities involving sublinear and superlinear terms. We shall impose no
growth restriction on the nonlinear term and consequently our problem can be
super-critical by means of Sobolev spaces.
