Elliptic variational problems with mixed nonlinearities

Abstract

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\ge 2$, as \begin{equation} \Lp u+u^{p-1}=λ\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min \{p^{\ast },q\}$ if $p<N$ while $p<r<q$ if $p\ge N$, and $\lambda >0$ is a parameter.