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

Abstract

In this work, we investigate the existence of the limit at infinity of weak solutions of the nonhomogeneous equation $-\mathrm{div}\left(\right|\nabla u{|}^{p-2}A\left(\right|\nabla u\left|\right)\nabla u)=f$ in the exterior domain $R^n﹨K$, where $K\subset R^n$ is a compact set. Indeed, for any $p\in (1,+\infty )$ and $n\ge 2$, we prove that the solutions converge at infinity if A satisfies some growth conditions and $f\in L^{\infty }\left(R^n\right)$ has some decay property. 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. In addition, for $p>n$, we show that for any continuous function ϕ defined on ∂K, the problem$\{\begin{array}{cc}-\mathrm{div}\left(\right|\nabla u{|}^{p-2}A\left(\right|\nabla u\left|\right)\nabla u)=f& \text{ in }{R}^n﹨K\\ u=\varphi ,& \text{ on }\partial K\end{array}$ has a bounded weak solution in $C\left(\overline{R^n﹨K}\right)\cap C^1(R^n﹨K)$, provided A and f are suitable. Furthermore, if $\varphi \in C^{\alpha }\left(K\right)$, then this solution is in $C^{\alpha }\left(R^n\right)$.