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Математические заметки, 2022, том 112, выпуск 1, статья опубликована в англоязычной версии журнала
(Mi mzm13275)
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Статьи, опубликованные в английской версии журнала
On Stable Solutions to a Weighted Degenerate
Elliptic Equation with Advection Terms
Dao Trong Quyeta, Dao Manh Thangb a Academy of Finance, Hanoi, Vietnam
b Hung Vuong High School for Gifted Students, Phu Tho, Vietnam
Аннотация:
In this paper, we study the elliptic equations
$$
-G_\alpha u+c({\rm x})\cdot\nabla_\alpha u=h({\rm x} )e^{u}, \qquad {\rm x} =
(x,y) \in \mathbb R^{N_{1}}\times \mathbb R^{N_{2}}=\mathbb R^{N},
$$
where
$G_{\alpha} =\Delta_{x}+ ( 1+\alpha )^{2}\lvert x\rvert^{2\alpha}\Delta_{y}$,
$\alpha > 0$, is the Grushin operator. Here, the advection term $c({\rm x})$
is a smooth, divergence free vector field satisfying certain decay condition and
$h({\rm x}) $ is a continuous function such that $h({\rm x} )\geq C|{\rm x}|^l$,
$l\geq 0$, where $|{\rm x}|$ is the Grushin norm of ${\rm x}$.
We will prove that the equation has no stable solutions provided that
$$
N_{\alpha}< 10+ 4 l,
$$
where $N_\alpha:=N_1+(1+\alpha)N_2$ is the homogeneous dimension of $\mathbb R^N$ associated to the Grushin operator.
Ключевые слова:
Liouville type theorems, Advection terms, Stable solutions, elliptic equations.
Поступило: 31.08.2021
Образец цитирования:
Dao Trong Quyet, Dao Manh Thang, “On Stable Solutions to a Weighted Degenerate
Elliptic Equation with Advection Terms”, Math. Notes, 112:1 (2022), 109–115
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/mzm13275
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