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} &-ε^2Δu+V(x)u=λu+f(u), \quad \quad \hbox{in }\mathbb{R}^N, &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}ε^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.