On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole RN Via Penalization Method

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems - ϵ 2 Δ u + V ( x ) u = λ u + f ( u ) , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, V : R N → [ 0 , ∞ ) 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 ).