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On the fractional Lazer-McKenna conjecture with critical growth
Published online by Cambridge University Press: 11 August 2021
Abstract
This paper deals with the following fractional elliptic equation with critical exponent
\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]
$\lambda$,
$\bar {\nu }\in {{\mathfrak R}}$,
$s\in (0,1)$,
$2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$,
$(-\Delta )^{s}$ is the fractional Laplace operator,
$\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and
$\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition
$u=0$ in
${{\mathfrak R}}^{N}\setminus \Omega$. Under suitable conditions on
$\lambda$ and
$\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as
$|\bar \nu | \to \infty$ .
MSC classification
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 879 - 911
- Copyright
- Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
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