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Some remarks on Liouville type theorems. (English) Zbl 1343.35049

Chipot, Michel (ed.) et al., Recent advances in nonlinear analysis. Proceedings of the international conference on nonlinear analysis, Hsinchu, Taiwan, November 20–25, 2006. Hackensack, NJ: World Scientific (ISBN 978-981-270-924-0/hbk). 43-65 (2008).
Summary: The goal of this note is to present elementary proofs of statements related to the Liouville theorem.
This note is divided as follows. In the next section we introduce an elementary estimate which is used later. In Section 3 we present some Liouville type results, i.e., we show that under some conditions on \(a\), \[ -\nabla\cdot (A(x)\nabla u(x)) + a(x)u(x) = 0\quad\text{ in }\mathcal D'(\mathbb R^k), \tag{1.2} \] where \(a\in L^1_{\text{loc}}(\mathbb R^k)\) and \(a\geq 0\), does not admit nontrivial bounded solutions. Finally in the last section we give an almost sharp criterion for the existence of nontrivial solutions.
For the entire collection see [Zbl 1141.35004].

MSC:

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J15 Second-order elliptic equations
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