We consider the p p -Laplacian problem \[ − d i v ( a ( x ) | ∇ u | p − 2 ∇ u ) + b ( x ) | u | p − 2 u = f ( x , u ) , x ∈ Ω , u | ∂ Ω = 0 , lim | x | → ∞ u = 0 , - div(a(x)|\nabla u{|^{p - 2}}\nabla u) + b(x)|u{|^{p - 2}}u = f(x,u),\quad x \in \Omega ,\quad u{|_{\partial \Omega }} = 0,\quad \lim \limits _{|x| \to \infty } u = 0, \] where 1 > p > n , Ω ( ⊂ R n ) 1 > p > n,\Omega ( \subset {R^n}) is an exterior domain. Under certain conditions, we show the existence of solutions for this problem via critical point theory.