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 $f$.
Categories: math.AP