TY - JOUR
T1 - Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
AU - Cingolani, S.
AU - Gallo, Marco
AU - Tanaka, K.
PY - 2021
Y1 - 2021
N2 - We study existence of solutions for the fractional problem
\begin{equation*}
(P_m) \quad \parag{
(-\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{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 Lagrangian 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 \emph{Pohozaev's mountain} in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in \cite{HT0,IT0}.
A multiplicity result of infinitely many normalized solutions is also obtained if $g$ is odd.
AB - We study existence of solutions for the fractional problem
\begin{equation*}
(P_m) \quad \parag{
(-\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{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 Lagrangian 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 \emph{Pohozaev's mountain} in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in \cite{HT0,IT0}.
A multiplicity result of infinitely many normalized solutions is also obtained if $g$ is odd.
KW - Pohozaev identity
KW - Radially symmetric solution
KW - Normalized solution
KW - Fractional Laplacian
KW - Nonlinear Schrödinger equation
KW - Prescribed mass
KW - Lagrange multiplier
KW - Pohozaev identity
KW - Radially symmetric solution
KW - Normalized solution
KW - Fractional Laplacian
KW - Nonlinear Schrödinger equation
KW - Prescribed mass
KW - Lagrange multiplier
UR - http://hdl.handle.net/10807/229085
UR - https://iopscience.iop.org/article/10.1088/1361-6544/ac0166/meta
U2 - 10.1088/1361-6544/ac0166
DO - 10.1088/1361-6544/ac0166
M3 - Article
SN - 0951-7715
VL - 34
SP - 4017
EP - 4056
JO - Nonlinearity
JF - Nonlinearity
ER -