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

Abstract

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.