论文标题
无均匀退化椭圆方程在外部领域的无限溶液中的收敛
Convergence at infinity for solutions of nonhomogeneous degenerate elliptic equations in exterior domains
论文作者
论文摘要
在这项工作中，首先我们证明，对于任何紧凑的集合$ k \ subset \ mathbb {r}^{n} $以及在$ \ partial k $上定义的任何连续函数$ ϕ $，存在$ c（\ bar {\ bar {\ mathb {r} r}^r}^{n}^{n}^{ c^1（\ Mathbb {r}^{n} \ BackSlash K）$ to to to to to to to to to to to to exterior dirichlet问题$$ begin {cases} - {\ rm div} \ big（\，| \ nabla u |^{p-2} a（\，| \ nabla u | \，）\ nabla u \，\ big）= f＆\ text {in} \; \; \; \; \; \; u = ϕ＆\ text {on} \ partial k \ end {cases} $$提供的$ p> n $，$ a $满足某些增长条件，而$ f \ in l^{\ infty}（\ mathbb {r}^n）$符合适当的衰减率。此后，我们获得了无限解决方案的限制，以解决此问题的任何$ p \ in（1，+\ infty）$和$ n \ geq2 $。此外，对于$ p> n $，我们可以证明该解决方案以某种速率汇合，对于$ p <n $，即使对于某些无限制的$ f $，也会汇聚。
In this work, first we prove that for any compact set $K\subset\mathbb{R}^{n}$ and any continuous function $ϕ$ 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 $$ \begin{cases} -{\rm div}\big(\,|\nabla u|^{p-2}A(\,|\nabla u|\,)\nabla u\,\big)=f & \text{ in }\, \mathbb{R}^n \backslash K \;\;\;\;\;\; u = ϕ& \text{ on } \partial K \end{cases} $$ 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 $f$.
