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Mat. Zametki, 1996, Volume 60, Issue 4, Pages 556–568 (Mi mz1862)  

Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain

A. B. Shapoval

International Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS

Abstract: We consider the solutions of the inequality $Lu\le\varphi(|{\operatorname{grad}u}|)$, where $L$ is a uniformly elliptic homogeneous operator and $\varphi$ is a function increasing faster than any linear function but not faster than $\xi\ln\xi$, in the unbounded domain
$$ \{x\in\mathbb R^n| \sum_{i=2}^nx_i^2<(\psi(x_1))^2, -\infty<x_1<\infty\}, $$
where $\psi$ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1}\frac{dr}{\psi(r)}$. For the special case in which $\varphi(\xi)=a\xi\ln\xi+C$, the solutions $u(x_1,x_2,…,x_n)$ grow as $(\int_0^{x_1}\frac{dr}{\varphi(r)})^N$, where $N$ is any given number and $a=a(N)$.

DOI: https://doi.org/10.4213/mzm1862

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English version:
Mathematical Notes, 1996, 60:4, 415–424

Bibliographic databases:

UDC: 517.9
Received: 14.07.1994

Citation: A. B. Shapoval, “Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain”, Mat. Zametki, 60:4 (1996), 556–568; Math. Notes, 60:4 (1996), 415–424

Citation in format AMSBIB
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\by A.~B.~Shapoval
\paper Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain
\jour Mat. Zametki
\yr 1996
\vol 60
\issue 4
\pages 556--568
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\crossref{https://doi.org/10.4213/mzm1862}
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\transl
\jour Math. Notes
\yr 1996
\vol 60
\issue 4
\pages 415--424
\crossref{https://doi.org/10.1007/BF02305424}
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