  In this paper, we study the existence and multiplicity results of nontrivial
positive solutions to the following quasilinear elliptic equation on $\RN$,
when $N\geq2$, \begin{equation} \Lp
u=\lambda\hspace{0.2mm}K(x)u^{r-1}-V(x)u^{q-1}.\nonumber \end{equation} Here,
$K(x),V(x)>0$ are suitable potentials, $1<p<r<q<\infty$, and $\lambda>0$ is a
parameter. To study this problem, some compact embedding results regarding
$\MVR\hookrightarrow\LKR$ are proved that unify and extend some recent results
of the author \cite{Ha1,Ha2,Ha3}.
