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

$$\begin{aligned} \left\{ \begin{aligned}&-\epsilon ^2\Delta u+V(x)u=\lambda u+f(u), \quad \quad \text {in }\mathbb {R}^N,\\&\int _{\mathbb {R}^{N}}|u|^{2}dx=a^{2}\epsilon ^N, \end{aligned} \right. \end{aligned}$$

where \(a,\epsilon >0\), \(\lambda \in \mathbb {R}\) is an unknown parameter that appears as a Lagrange multiplier, \(V:\mathbb {R}^N \rightarrow [0,\infty )\) 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 (Calc. Var. Partial Differential Equations 4, 121–137 1996).