On the fractional Lazer-McKenna conjecture with critical growth
| Autor: | Qi Li, Shuangjie Peng |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 152:879-911 |
| ISSN: | 1473-7124 0308-2105 |
| DOI: | 10.1017/prm.2021.40 |
| Popis: | 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 $|\bar \nu | \to \infty$ . |
| Databáze: | OpenAIRE |
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