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 $\varphi$ defined on $\partial K$, there exists a bounded weak solution in $C(\overline{{\mathbb{R}}^n\backslash K})\cap C^1({\mathbb{R}}^n\backslash K)$ to the exterior Dirichlet problem $$\{\begin{array}{llc}-\mathrm{div}(|\nabla u{|}^{p-2}A(|\nabla u|)\nabla u)=f& \text{ in }{\mathbb{R}}^n\backslash Ku=\varphi & \text{ on }\partial K\end{array}$$ 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 $n\ge 2$. 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 $f$.