Note on normalized solutions to a kind of fractional Schrödinger equation with a critical nonlinearity

Source abstract

&lt;p&gt;In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity&lt;/p&gt;&lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{lll} (-\Delta)^su = \lambda u+|u|^{p-2}u+|u|^{2^\ast_s-2}u, &amp;amp; x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}u^2{\rm d}x = a^2, \ u\in H^{s}(\mathbb{R}^N), \end{array}\right. \end{eqnarray*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p&gt;where $N\geq2$, $s\in(0, 1)$, $a > 0$, $2 < p < 2^\ast_s\triangleq\frac{2N}{N-2s}$ and $(-\Delta)^s$ is the fractional Laplace operator. In the purely $L^2$-subcritical perturbation case $2 < p < 2+\frac{4s}{N}$, we prove the existence of a second normalized solution under some conditions on $a$, $p$, $s$, and $N$. This is a continuation of our previous work (&lt;italic&gt;Z. Angew. Math. Phys.&lt;/italic&gt;, &lt;bold&gt;73&lt;/bold&gt; (2022) 149) where only one solution is obtained.&lt;/p&gt;