  In this paper we study the existence of multiple normalized solutions to the
following class of elliptic problems \begin{align*}
  \left\{
  \begin{aligned}
  &-\epsilon^2\Delta u+V(x)u=\lambda u+f(u), \quad
  \quad
  \hbox{in }\mathbb{R}^N,
  &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}\epsilon^N,
  \end{aligned}
  \right. \end{align*} where $a,\epsilon>0$, $\lambda\in \mathbb{R}$ is an
unknown parameter that appears as a Lagrange multiplier, $V:\mathbb{R}^N \to
[0,\infty)$ is a continuous function, and $f$ is a continuous function with
$L^2$-subcritical growth. It is
  proved that the number of normalized solutions is related to the topological
richness of the set where the potential $V$ attains its minimum value. In the
proof of our main result, we apply minimization techniques,
Lusternik-Schnirelmann category and the penalization method due to del Pino and
Felmer.
