Project Euclid

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: $$ \begin{cases} -\epsilon^{p}\Delta_{p}{u}+(\lambda A(x)+1)|u|^{p-2}u=f(u), \,\,\, \mathbb{R}^{N}\\ u(x)>0 \,\,\, \mbox{in} \,\, \mathbb{R}^{N}, \end{cases} $$ where $\Delta_{p}$ is the p-Laplacian operator, $ N >p \geq 2$, $\lambda$ and $\epsilon$ are positive parameters, $A$ is a nonnegative continuous function and $f$ is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of ${\rm{int}}A^{-1}(0)$ for all sufficiently large $\lambda$ and small~$\epsilon$.