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On the fractional Lazer-McKenna conjecture with critical growth

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  11 August 2021

Qi Li  and
Shuangjie Peng
Qi Li
Affiliation:
College of Science, Wuhan University of Science and Technology, Wuhan 430065, People's Republic of China School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People's Republic of China (qili@mails.ccnu.edu.cn)
Shuangjie Peng
Affiliation:
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, People's Republic of China (sjpeng@mail.ccnu.edu.cn)
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Abstract

This paper deals with the following fractional elliptic equation with critical exponent

\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]
where $\lambda$, $\bar {\nu }\in {{\mathfrak R}}$, $s\in (0,1)$, $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$, $(-\Delta )^{s}$ 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 $|\bar \nu | \to \infty$ .

Keywords

fractional Laplacefirst eigenfunctionLyapunov-Schmidt reduction

MSC classification

Primary: 35A01: Existence problems
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 879 - 911
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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