Compact Sobolev embeddings and positive solutions to a quasilinear equation with indefinite nonlinearities

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}.