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

Abstract

<p>In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{lll} (-\Delta)^su = \lambda u+|u|^{p-2}u+|u|^{2^\ast_s-2}u, &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} </tex-math></disp-formula></p><p>where $ N\geq2 $, $ s\in(0, 1) $, $ a &gt; 0 $, $ 2 &lt; p &lt; 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 &lt; p &lt; 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 (<italic>Z. Angew. Math. Phys.</italic>, <bold>73</bold> (2022) 149) where only one solution is obtained.</p>