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Труды Математического института имени В. А. Стеклова, 2002, том 236, страницы 447–461
(Mi tm314)
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On the Asymptotic Behavior of Solutions
of a Semilinear Elliptic Boundary Problem in Unbounded Domains
Yu. V. Egorova, V. A. Kondrat'evb a Université Paul Sabatier
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Аннотация:
We consider solutions of an elliptic linear equation $Lu=0$ of second order
in an unbounded domain $Q$ in $\mathbb R^n$ supposing that
$Q\subset\{x=(x',x_n)\colon 0<x_n<\infty,\, |x'|<\gamma(x_n)\}$, where
$1\le \gamma(t)\le At+B$, and that $u$ satisfies the nonlinear boundary
condition $\frac{\partial u}{\partial N}+k(x)u+b(x)|u(x)|^{p-1}u(x)=0$ on
the part of the boundary of $Q$ where $x_n>0$. We show that any such
solution $u$ growing moderately at infinity tends to $0$ as $|x|\to\infty$.
Earlier we showed this theorem for the case $\gamma(x_n)=B$, i.e. for a cylindrical domain $Q=\Omega\times (0,\infty)$, $\Omega\subset\mathbb R^{n-1}$, and for the case when $A\le A_0$ with a constant $A_0$ sufficiently small. Here we admit any value of $A_0$. Our theorem is true even for the domain which is an outer part of a cone, and for the
half-space $x_n>0$. Besides, we consider here more general operators $L$
with lower order terms. Notice that the new proof is quite different from
those in our earlier works.
Поступило в феврале 2001 г.
Образец цитирования:
Yu. V. Egorov, V. A. Kondrat'ev, “On the Asymptotic Behavior of Solutions
of a Semilinear Elliptic Boundary Problem in Unbounded Domains”, Дифференциальные уравнения и динамические системы, Сборник статей. К 80-летию со дня рождения академика Евгения Фроловича Мищенко, Труды МИАН, 236, Наука, МАИК «Наука/Интерпериодика», М., 2002, 447–461; Proc. Steklov Inst. Math., 236 (2002), 434–448
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/tm314 https://www.mathnet.ru/rus/tm/v236/p447
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