| Abstract: |
| In this talk, we consider the following class of fractional relativistic Schr\odinger equations:
$\begin{equation*}
\left\{
\begin{array}{ll}
(-\varepsilon^{2}\Delta+m^{2})^{s}u + V(x) u= f(u)+\gamma u^{2^{*}_{s}-1} \mbox{ in } \mathbb{R}^{N}, \
u \in H^{s}(\mathbb{R}^{N}), \quad u>0 \, \mbox{ in } \mathbb{R}^{N}, \
\end{array}
\right.
\end{equation*}$
where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $m>0$, $N> 2s$, $\gamma\in \{0, 1\}$, and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent. Here, the pseudo-differential operator $(-\varepsilon^{2}\Delta+m^{2})^{s}$ is simply defined in Fourier variables by the symbol $(\varepsilon^{2}|\xi|^{2}+m^{2})^{s}$, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Utilizing the extension method and penalization techniques, we first show that there exists a family of positive solutions $u_{\varepsilon}\in H^{s}(\mathbb{R}^{N})$, with exponential decay, that concentrate around a local minimum of $V$ as $\varepsilon\rightarrow 0$. Finally, we combine the generalized Nehari manifold method with the Ljusternik-Schnirelman theory to relate the number of positive solutions to the topology of the set where the potential $V$ attains its minimum value. |
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