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

Qi Han

doi:10.1016/j.jmaa.2019.04.062

paper

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

arXiv:1901.02526 · doi:10.1016/j.jmaa.2019.04.062

Abstract

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on $\RN$, when $N \geq 2$ , \begin{equation} \Lp u=λ\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 $λ > 0$ 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}.