| Abstract: |
| We consider the following class of fractional relativistic Schr\odinger equations:
\begin{equation*}
\left\{
\begin{array}{ll}
(-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\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, $s\in (0, 1)$, $m>0$, $N> 2s$,
$(-\Delta+m^{2})^{s}$ is the fractional relativistic Schr\odinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous subcritical nonlinearity.
We first show that, for $\varepsilon>0$ small enough, the above problem has a weak solution $u_{\varepsilon}$ (with exponential decay at infinity) which concentrates around a local minimum point of $V$ as $\varepsilon\rightarrow 0$.
We also relate the number of positive solutions with the topology of the set where the potential $V$ attains its minimum value.
The main results will be established by using a penalization technique,
the generalized Nehari manifold method and Ljusternik-Schnirelman theory. |
|