  In this work, first we prove that for any compact set
$K\subset\mathbb{R}^{n}$ and any continuous function $\phi$ defined on
$\partial K$, there exists a bounded weak solution in
$C(\bar{\mathbb{R}^{n}\backslash K}) \cap C^1(\mathbb{R}^{n}\backslash K)$ to
the exterior Dirichlet problem $$ \begin{cases}
  -{\rm div}\big(\,|\nabla u|^{p-2}A(\,|\nabla u|\,)\nabla u\,\big)=f & \text{
in }\, \mathbb{R}^n \backslash K
  \;\;\;\;\;\; u = \phi & \text{ on } \partial K
  \end{cases}
  $$ provided $p > n$, $A$ satisfies some growth conditions and $f\in
L^{\infty}(\mathbb{R}^n)$ meets a suitable pointwise decay rate. We obtain
thereafter the existence of the limit at the infinity for solutions to this
problem, for any $p\in(1,+\infty)$ and $n\geq2$. Moreover, for $p > n$ we can
show that the solutions converge at some rate and for $p <n$ the convergence
holds even for some unbounded $f$.
