Existence and multiplicity of solutions for a class of quasilinear elliptic field equation on mathbb{R}^(N)

In this paper, we establish existence and multiplicity of solutions for the following class of quasilinear field equation $$ -\Delta u+V(x)u-\Delta_{p}u+W'(u)=0, \,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, \eqno{(P)} $$ where $u=(u_1,u_2,...,u_{N+1})$, $p>N \geq 2$, $W$ is a singular function and $V$ is a positive continuous function.