Normalized solutions for Sobolev critical fractional Schrödinger equation

Abstract

Abstract In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: ( − Δ ) s u + λ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R N , ( P m ) ∫ R N ∣ u ∣ 2 d x = m 2 , \hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" voffset="-2.9ex">\left({P}_{m})</mml:mpadded>\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , N ≥ 2 N\ge 2 , m > 0 m\gt 0 , 2 s * ≔ 2 N N − 2 s {2}_{s}^{* }:= \frac{2N}{N-2s} , λ \lambda is an unknown parameter that will appear as a Lagrange multiplier, and f f is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about f f , with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to ( P m ) \left({P}_{m}) . We mainly extend the results of Appolloni and Secchi (Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu (A mass supercritical problem revisited, Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave (Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610), when s = 1 s=1 .