Existence and limit behavior of prescribed L 2 ‐norm solutions for Schrödinger‐Poisson‐Slater systems in R 3

In this paper, we study constraint minimizers of the following L 2 −critical minimization problem:e ( N ) : = inf { E ( u ) , u ∈ H 1 ( R 3 )and∫ R 3| u | 2 d x = N > 0 } , where E ( u ) is the Schrödinger‐Poisson‐Slater functionalE ( u ) : = ∫ R 3| ∇ u | 2 d x − 1 2 ∫ R 3∫ R 3u 2 ( y ) u 2 ( x ) | x − y | d y d x − 3 5 ∫ R 3m ( x ) | u | 10 3d x , and N denotes the mass of the particles in the Schrödinger‐Poisson‐Slater system. We prove that e ( N ) admits minimizers for N < N ∗ : = ‖ Q ‖ 2 2and, however, no minimizers for N > N ∗ , where Q ( x ) is the unique positive solution of △ u − u + u 7 3= 0 in R 3 . Some results on the existence and nonexistence of minimizers for e ( N ∗ ) are also established. Further, when e ( N ∗ ) does not admit minimizers, the limit behavior of minimizers as N ↗ N ∗ is also analyzed rigorously.