Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-06-30 , DOI: 10.1007/s00033-020-01338-0 Zongming Guo , Xia Huang , Dong Ye
We consider positive solutions to the weighted elliptic problem
$$\begin{aligned} -\text{ div } (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\text{ in } {\mathbb {R}}^N \backslash {{\overline{B}}},\quad u=0 \;\; \text{ on } \partial B, \end{aligned}$$where B is the standard unit ball of \({\mathbb {R}}^N\). We give a complete answer for the existence question for \(N':=N+\theta >2\) and \(p > 0\). In particular, for \(N' > 2\) and \(\tau :=\ell -\theta >-2\), it is shown that for \(0< p \le p_s:=\frac{N'+2+2 \tau }{N'-2}\), the only nonnegative solution to the problem is \(u \equiv 0\). This nonexistence result is new, even for the classical case \(\theta = \ell = 0\) and \(\frac{N}{N-2} < p \le \frac{N+2}{N-2}\), \(N \ge 3\). The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption.
中文翻译：
外部域中加权椭圆方程的存在与不存在结果
我们考虑加权椭圆问题的正解
$$ \ begin {aligned}-\ text {div}（| x | ^ \ theta \ nabla u）= | x | ^ \ ell u ^ p \; \; \ text {in} {\ mathbb {R}} ^ N \反斜杠{{\ overline {B}}}，\ quad u = 0 \; \; \ text {on} \ partial B，\ end {aligned} $$其中B是\（{\ mathbb {R}} ^ N \）的标准单位球。我们给出\（N'：= N + \ theta> 2 \）和\（p> 0 \）的存在问题的完整答案。特别是，对于\（N'> 2 \）和\（\ tau：= \ ell-\ theta> -2 \），对于\（0 <p \ le p_s：= \ frac {N' + 2 + 2 \ tau} {N'-2} \），该问题的唯一非负解是\（u \ equiv 0 \）。这个不存在的结果是新的，即使对于经典情况\（\ theta = \ ell = 0 \）和\（\ frac {N} {N-2} <p \ le \ frac {N + 2} {N-2 } \），\（N \ ge 3 \）。这里有趣的特征是我们不需要无穷大的任何行为或任何对称假设。
京公网安备 11010802027423号