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

TL;DR

This paper proves the existence and convergence at infinity of solutions to a class of nonhomogeneous degenerate elliptic equations in exterior domains, with detailed results depending on the parameters p and n.

Contribution

It establishes the existence of bounded solutions with prescribed boundary data and analyzes their convergence at infinity under various conditions on p, n, and the source term.

Findings

Solutions exist for p > n with prescribed boundary conditions.

Solutions converge at infinity with specific rates for p > n.

Convergence at infinity also holds for some unbounded source terms when p < n.

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 $f$.