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Existence of positive solutions for Kirchhoff-type problem in exterior domains
Published online by Cambridge University Press: 04 April 2023
Abstract
We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$:(*)
\begin{align}\left\{\begin{array}{ll}-\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\\\u=0, & x\in\partial \Omega,\\\end{array}\right.\end{align}
$b\geq0$, and λ > 0 are constants,
$\partial\Omega\neq\emptyset$,
$\mathbb{R}^{3}\backslash\Omega$ is bounded,
$u\in H_{0}^{1}(\Omega)$, and
$f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if
\begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}
$H^1(\mathbb R^3)$ and the diameter of the hole
$\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.
- Type
- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 1 , February 2023 , pp. 182 - 217
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.
References
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