Abstract

Nonexistence of nontrivial nonnegative classical solutions is obtained for the problems: 0.1$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=u^p \;\;\; &{}\text{ in } {\mathbb {R}}^N \backslash {\overline{B}},\\ u=\Delta u=0 \;\;\; &{}\text{ on } \partial B \end{array} \right. \end{aligned}$$with \(1<p\le \frac{N+4}{N-4}\), and 0.2$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=u^p \;\;\; &{}\text{ in } {\mathbb {R}}^N \backslash {\overline{B}},\\ u=\frac{\partial u}{\partial \nu }=0 \;\;\; &{}\text{ on } \partial B, \end{array} \right. \end{aligned}$$where \(1<p<\frac{N+4}{N-4}\), \(B \subset {\mathbb {R}}^N \; (N \ge 5)\) is the unit ball, \(\nu \) is the unit outward normal vector of \(\partial B\) relative to B. The interesting features in our proof are that neither asymptotic behavior of u at infinity nor symmetric property of u are required. Moreover, when \(p=\frac{N+4}{N-4}\), we can also obtain nonexistence of nontrivial nonnegative classical radial solutions of (0.2). Nonexistence of nontrivial nonnegative classical solutions without symmetry property of (0.2) with \(p=\frac{N+4}{N-4}\) is still open. It is well known that problems (0.1) and (0.2) admit a unique positive radial solution \(u \in C^4 (\mathbb {R}^N \backslash B)\) for \(p>\frac{N+4}{N-4}\) respectively.

中文翻译：

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外域半线性双调和问题的Liouville型结果

摘要

对于这些问题，获得了非平凡非负经典解的不存在： 0.1$$ \ begin {aligned} \ left \ {\ begin {array} {ll} \ Delta ^ 2 u = u ^ p \; \; \; ＆{} \ text {in} {\ mathbb {R}} ^ N \反斜杠{\ overline {B}}，\\ u = \ Delta u = 0 \; \; \; ＆{} \ text {on} \ partial B \ end {array} \ right。\ end {aligned} $$与\（1 <p \ le \ frac {N + 4} {N-4} \），和0.2$$ \ begin {aligned} \ left \ {\ begin {array} {ll} \ Delta ^ 2 u = u ^ p \; \; \; ＆{} \ text {in} {\ mathbb {R}} ^ N \反斜杠{\ overline {B}}，\\ u = \ frac {\ partial u} {\ partial \ nu} = 0 \; \; \; ＆{} \ text {on} \ partial B，\ end {array} \ right。\ end {aligned} $$其中\（1 <p <\ frac {N + 4} {N-4} \）， \（B \ subset {\ mathbb {R}} ^ N \;（N \ ge 5 ）\）是单位球， \（\ nu \）是\（\ partial B \）相对于B的单位向外法向向量。我们的证明中有趣的特征是，既不需要u在无穷远处的渐近行为，也不需要u的对称性质。此外，当\（p = \ frac {N + 4} {N-4} \），我们还可以获得（0.2）的非平凡非负经典径向解的不存在。没有对称性为（0.2）与\（p = \ frac {N + 4} {N-4} \）的非平凡非负经典解的不存在仍然是开放的。众所周知的问题（0.1）和（0.2）接纳一个唯一的正径向溶液\（U \用C ^ 4（\ mathbb {R} ^ N \反斜杠B）\）为\（P> \压裂{N +4} {N-4} \）。