Convergence at infinity for solutions of nonhomogeneous degenerate elliptic equations in exterior domains

Abstract

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 $f$.