We prove existence of infinitely many solutions $u \in H^1_r(\mathbb{R}^N)$ for the nonlinear Choquard equation $$- \Delta u + \mu u =(I_\alpha*F(u)) f(u) \quad \hbox{in}\ \mathbb{R}^N,$$ where $N\geq 3$, $\alpha\in (0,N)$, $I_\alpha(x) := \frac{\Gamma(\frac{N-\alpha}{2})}{\Gamma(\frac{\alpha}{2}) \pi^{N/2} 2^\alpha } \frac{1}{|x|^{N- \alpha}}$, $x \in \mathbb{R}^N \setminus\{0\}$ is the Riesz potential, and $F$ is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: $\mu$ is a fixed positive constant or $\mu$ is unknown and the $L^2$-norm of the solution is prescribed, i.e. $\int_{\mathbb{R}^N} |u|^2 =m&gt;0$. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions in \cite{BL2}, we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results in \cite{MS2}, due to Moroz and Van Schaftingen.

Cingolani, S., Gallo, M., Tanaka, K., Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, <<CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS>>, 2022; 61 (2): 1-34. [doi:10.1007/s00526-021-02182-4] [https://hdl.handle.net/10807/227436]

### Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities

#### Abstract

We prove existence of infinitely many solutions $u \in H^1_r(\mathbb{R}^N)$ for the nonlinear Choquard equation $$- \Delta u + \mu u =(I_\alpha*F(u)) f(u) \quad \hbox{in}\ \mathbb{R}^N,$$ where $N\geq 3$, $\alpha\in (0,N)$, $I_\alpha(x) := \frac{\Gamma(\frac{N-\alpha}{2})}{\Gamma(\frac{\alpha}{2}) \pi^{N/2} 2^\alpha } \frac{1}{|x|^{N- \alpha}}$, $x \in \mathbb{R}^N \setminus\{0\}$ is the Riesz potential, and $F$ is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: $\mu$ is a fixed positive constant or $\mu$ is unknown and the $L^2$-norm of the solution is prescribed, i.e. $\int_{\mathbb{R}^N} |u|^2 =m>0$. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions in \cite{BL2}, we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results in \cite{MS2}, due to Moroz and Van Schaftingen.
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2022
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Cingolani, S., Gallo, M., Tanaka, K., Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, <<CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS>>, 2022; 61 (2): 1-34. [doi:10.1007/s00526-021-02182-4] [https://hdl.handle.net/10807/227436]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/227436
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