| Abstract: |
| In this talk, I will focus on the existence of $2\pi$-periodic solutions to the following fractional critical problem:
\begin{equation*}
\left\{
\begin{array}{ll}
(-\Delta+m^{2})^{s}u-m^{2s}u= f(x, u)+W(x) |u|^{2^{*}_{s}-2}u &\mbox{ in } (-\pi,\pi)^{N}, \
u(x+2\pi e_{i})=u(x) \mbox{ for all } x \in \mathbb{R}^{N}, i=1, \dots, N,
\end{array}
\right.
\end{equation*}
where $s\in (0,1)$, $m\geq 0$, $N\geq 4s$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, $W(x)$ is a $2\pi$-periodic positive continuous function, and $f(x, u)$ is a superlinear $2\pi$-periodic (in $x$) continuous function with subcritical growth and $(e_{i})$ is the canonical basis in $\mathbb{R}^{N}$. When $m>0$, I will show the existence of a nonconstant periodic solution by using the extension method in periodic setting and applying the Linking theorem. The case $m=0$ will be studied by means of a limit procedure. |
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