| Abstract: |
| In this talk, I will consider the following doubly nonlocal nonlinear elliptic problem:
$$
\left\{
\begin{array}{ll}
(-\Delta+m^{2})^{s}u+\omega u=(I_{\alpha}*F(u)) F`(u) \mbox{ in } \mathbb{R}^{N},
\
u\in H^{s}(\mathbb{R}^{N}),
\end{array}
\right.
$$
where $N\geq 2$, $s\in (0, 1)$, $m>0$, and $\omega>-m^{2s}$. Here, $(-\Delta+m^{2})^{s}$ denotes the fractional relativistic Schr\odinger operator, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0, N)$, and $F:\mathbb{R}\to \mathbb{R}$ is a $C^1$-nonlinearity of Berestycki--Lions type.
I will discuss the existence of least energy solutions, as well as their qualitative properties, including regularity, decay, sign, and symmetry. |
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