Annali di Matematica Pura ed Applicata ( IF 1.0 ) Pub Date : 2020-05-02 , DOI: 10.1007/s10231-020-00987-2 A. Aghajani , C. Cowan
We prove that if the elliptic problem \(-\Delta u+b(x)|\nabla u|=c(x)u\) with \(c\ge 0\) has a positive supersolution in a domain \(\varOmega \) of \( {\mathrm {R}}^{N\ge 3}\), then c, b must satisfy the inequality
$$\begin{aligned} \sqrt{ \int _\varOmega c\phi ^2}\le \sqrt{ \int _\varOmega | \nabla \phi |^2}+\sqrt{ \int _\varOmega \frac{b^2}{4}\phi ^2},\quad \phi \in C_c^\infty (\varOmega ). \end{aligned}$$As an application, we obtain Liouville-type theorems for positive supersolutions in exterior domains when \(c(x)-\frac{b^2(x)}{4}>0\) for large |x|, but unlike the known results, we allow the case \(\lim _{|x|\rightarrow \infty }c(x)-\frac{b^2(x)}{4}=0\). The weights b and c are allowed to be unbounded. In particular, among other things, we show that if \(\tau :=\limsup _{|x| \rightarrow \infty }|xb(x)|<\infty, \) then this problem does not admit any positive supersolution if
$$\begin{aligned} \liminf _{|x| \rightarrow \infty }|x|^2c(x)> \frac{(N-2+\tau )^2}{4}, \end{aligned}$$and, when \(\tau =\infty , \) we have the same if
$$\begin{aligned} \limsup _{R\rightarrow \infty } R\left( \frac{ \inf _{R<|x|<2 R} (c(x)-\frac{b(x)^2}{4})}{\sup _{\frac{R}{2}<|x|<4 R}|b(x)|}\right) =\infty . \end{aligned}$$中文翻译：
关于不存在梯度项的椭圆型方程正解的不存在的注记
我们证明如果椭圆问题\（-\ Delta u + b（x）| \ nabla u | = c（x）u \）与\（c \ ge 0 \）在一个域\（\ varOmega \）的\（{\ mathrm {R}} ^ {N \ GE 3} \） ，然后ç， b必须满足不等式
$$ \ begin {aligned} \ sqrt {\ int _ \ varOmega c \ phi ^ 2} \ le \ sqrt {\ int _ \ varOmega | \ nabla \ phi | ^ 2} + \ sqrt {\ int _ \ varOmega \ frac {b ^ 2} {4} \ phi ^ 2}，\ quad \ phi \ in C_c ^ \ infty（\ varOmega）。\ end {aligned} $$作为应用，当\（c（x）-\ frac {b ^ 2（x）} {4}> 0 \）大时| | | | | | | | | | | | | | | | | | | | | | | x |，但不同于已知结果，我们允许情况为\（\ lim _ {| x | \ rightarrow \ infty} c（x）-\ frac {b ^ 2（x）} {4} = 0 \）。权重b和c可以不受限制。特别地，除其他外，我们证明，如果\（\ tau：= \ limsup _ {| x | \ rightarrow \ infty} | xb（x）| <\ infty，\），那么此问题将不接受任何正解如果
$$ \ begin {aligned} \ liminf _ {| x | \ rightarrow \ infty} | x | ^ 2c（x）> \ frac {（N-2 + \ tau）^ 2} {4}，\ end {aligned} $$并且，当\（\ tau = \ infty，\）时，如果
$$ \ begin {aligned} \ limsup _ {R \ rightarrow \ infty} R \ left（\ frac {\ inf _ {R <| x | <2 R}（c（x）-\ frac {b（x） ^ 2} {4}）} {\ sup _ {\ frac {R} {2} <| x | <4 R} | b（x）|} \ right）= \ infty。\ end {aligned} $$
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