Soliton Solution for Quasilinear Elliptic Equations Involving the $p$-Laplacian in $R ^N$

In this paper we use the variational methods, more precisely, the Mountain-Pass Theorem to obtain existence results for the following class of quasilinear elliptic problems:\[\begin{array}{lll}-\Delta_p u -\Delta_p (u^2)u +k V(x)|u|^{p-2} u= h(u), & \;\text{in}\; &\mathbb{R}^N,\\\end{array}\]where $ \Delta_p u := div(|\nabla u |^{p-2} \nabla u)$ is the $p$-Laplacian operator and $1