Article contents
Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1
Published online by Cambridge University Press: 26 January 2023
Abstract
We study the asymptotic behaviour, as $p\to 1^+$, of the solutions of the following inhomogeneous Robin boundary value problem:P
is a bounded domain in $\mathbb {R}^{N}$
with sufficiently smooth boundary, $\nu$
is its unit outward normal vector and $\Delta _p v$
is the $p$
-Laplacian operator with $p>1$
. The data $f\in L^{N,\infty }(\Omega )$
(which denotes the Marcinkiewicz space) and $\lambda,\,g$
are bounded functions defined on $\partial \Omega$
with $\lambda \ge 0$
. We find the threshold below which the family of $p$
–solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$
-Laplacian problem formally arising by taking $p\to 1^+$
in (P).
MSC classification
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 105 - 130
- Copyright
- Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
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