Multiplicity of positive solutions for a class of quasilinear problems

Abstract

In this paper we prove the existence and multiplicity of nontrivial weak solutions for quasilinear elliptic equations of the form $-L_p u +V(x)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$, where $L_p u\doteq \epsilon^{p}\Delta_p u +\epsilon^{p}\Delta_p (u^2)u$ and $V$ is a positive continuous potential bounded away from zero satisfying some conditions and the nonlinear term $h(u)$ has a subcritical growth type. Here, we use a variational method to get the multiplicity of positive solutions involving the Lusternick-Schnirelman category of the set where $V$ achieves its minimum value.