Normalized Solutions for the Critical Schrödinger–Bopp–Podolsky System

In this paper, we study the following Schrödinger—Bopp-Podolsky equation with prescribed mass - Δ u + λ u - κ 1 - e - | x | a | x | ∗ | u | 2 u = | u | 4 u , in R 3 , u > 0 , ∫ R 3 u 2 d x = c 2 , where c , a > 0 and κ ∈ R \ { 0 } are fixed constants and λ ∈ R appears as a Lagrange multiplier. For κ > 0 , using a truncation argument and a measure representation lemma due to Lions, we prove that the above system admits at least n pairs of radial normalized solutions u j a ( j = 1 , 2 , … , n ) with negative energy. Moreover, for each j = 1 , 2 , … , n , we study the asymptotic behavior of u j a with respect to a . For κ < 0 , we obtain a non-existence result by using a Liouville-type Theorem and the Pohozaev identity.