*Published Paper*

**Inserted:** 10 may 2022

**Last Updated:** 13 nov 2022

**Journal:** Nonlinearity

**Volume:** 34

**Number:** 6

**Pages:** 4017-4056

**Year:** 2021

**Doi:** 10.1088/1361-6544/ac0166

**Abstract:**

We study existence of solutions for the fractional problem \begin{equation**}
(P _{m)} \quad \left \{
\begin{aligned}
(-\Delta)^{{s}} u + \mu u &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr
\int_{{\mathbb{R}}^{N}} u^{2} dx &= m, & \cr
u \in H^{s}_{r&}(\mathbb{R}^{N),} &
\end{aligned}
\right. \label{problemx} \end{equation**} where $N\geq 2$, $s\in (0,1)$,
$m>0$, $\mu$ is an unknown Lagrange multiplier and $g \in C(\mathbb{R},
\mathbb{R})$ satisfies Berestycki-Lions type conditions. Using a Lagrange
formulation of the problem $(P_m)$, we prove the existence of a weak solution
with prescribed mass when $g$ has $L^2$ subcritical growth. The approach relies
on the construction of a minimax structure, by means of a Pohozaev's mountain
in a product space and some deformation arguments under a new version of the
Palais-Smale condition introduced in 21,25. A multiplicity result of
infinitely many normalized solutions is also obtained if $g$ is odd.

**Keywords:**
fractional Laplacian, Pohozaev identity, nonlinear SchrÃ¶dinger equation, Radially symmetric solution, Normalized solution, Lagrange multiplier, Prescribed mass