Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
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| Natura: | Preprint |
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2023
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| author | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| author_facet | Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga |
| contents | In this paper we study the following nonlinear fractional Choquard-Pekar equation \begin{equation}\label{eq_abstract} (-Δ)^s u + μu =(I_α*F(u)) F'(u) \quad \hbox{in}\ \mathbb{R}^N, \tag{$*$} \end{equation} where $μ>0$, $s \in (0,1)$, $N \geq 2$, $α\in (0,N)$, $I_α\sim \frac{1}{|x|^{N-α}}$ is the Riesz potential, and $F$ is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions $u \in H^s(\mathbb{R}^N)$, by assuming $F$ odd or even: we consider both the case $μ>0$ fixed and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed. Here we also simplify some arguments developed for $s=1$ in [Calc. Var. PDEs, 2022].
A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions [ARMA, 1983]; for \eqref{eq_abstract} the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as $μ$ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any $m>0$.
The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a $C^1$-regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_14003 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities Cingolani, Silvia Gallo, Marco Tanaka, Kazunaga Analysis of PDEs 35A01, 35A15, 35B06, 35B38, 35D30, 35J15, 35J20, 35J61, 35Q40, 35Q55, 35Q60, 35Q70, 35Q75, 35Q85, 35Q92, 35R09, 35R11, 45K05, 47G10, 47J30, 49J35, 58E05, 58J05 In this paper we study the following nonlinear fractional Choquard-Pekar equation \begin{equation}\label{eq_abstract} (-Δ)^s u + μu =(I_α*F(u)) F'(u) \quad \hbox{in}\ \mathbb{R}^N, \tag{$*$} \end{equation} where $μ>0$, $s \in (0,1)$, $N \geq 2$, $α\in (0,N)$, $I_α\sim \frac{1}{|x|^{N-α}}$ is the Riesz potential, and $F$ is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions $u \in H^s(\mathbb{R}^N)$, by assuming $F$ odd or even: we consider both the case $μ>0$ fixed and the case $\int_{\mathbb{R}^N} u^2 =m>0$ prescribed. Here we also simplify some arguments developed for $s=1$ in [Calc. Var. PDEs, 2022]. A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions [ARMA, 1983]; for \eqref{eq_abstract} the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as $μ$ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any $m>0$. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a $C^1$-regularity. |
| title | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities |
| topic | Analysis of PDEs 35A01, 35A15, 35B06, 35B38, 35D30, 35J15, 35J20, 35J61, 35Q40, 35Q55, 35Q60, 35Q70, 35Q75, 35Q85, 35Q92, 35R09, 35R11, 45K05, 47G10, 47J30, 49J35, 58E05, 58J05 |
| url | https://arxiv.org/abs/2305.14003 |