Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866908415151833088 |
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| author | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| author_facet | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| contents | We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \left \{
\begin{aligned}
(-Δ)^{s} u + μu &=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 $N\geq 2$, $s\in (0,1)$, $m>0$, $μ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_10747 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga Analysis of PDEs 35Q55, 35R11, 35J20, 58E05 We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \left \{ \begin{aligned} (-Δ)^{s} u + μu &=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 $N\geq 2$, $s\in (0,1)$, $m>0$, $μ$ 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. |
| title | Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation |
| topic | Analysis of PDEs 35Q55, 35R11, 35J20, 58E05 |
| url | https://arxiv.org/abs/2103.10747 |