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In this paper we study the existence of multiple normalized solutions to thefollowing 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 anunknown 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 topologicalrichness of the set where the potential $V$ attains its minimum value. In theproof of our main result, we apply minimization techniques,Lusternik-Schnirelmann category and the penalization method due to del Pino andFelmer.

On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^N$ via penalization method