Abstract:

In this article, we study constrained minimizers of the following variational problem
e(ρ):=???20180223??? E(u), ρ > 0,
where E(u) is the Schrödinger-Poisson-Slater (SPS) energy functional
E(u):=1/2 ∫_R³|▽u(x)|²dx -1/4 ∫_R³ ∫_R³ (u²|(y)u²(x))/(|x-y|)dydx -1/p ∫_R³|u(x)|^pdx in R³,
and p ∈ (2, 6). We prove the existence of minimizers for the cases 2< p < 10/3, ρ > 0, and p=10/3, 0< ρ < ρ^*, and show that e(ρ)=-∞ for the other cases, where ρ^*=||φ||₂² and φ(x) is the unique (up to translations) positive radially symmetric solution of -△u + u=u^7/3 in R³. Moreover, when e(ρ^*)=-∞, the blow-up behavior of minimizers as ρ ↗ ρ^* is also analyzed rigorously.

Key words: Schrödinger-Poisson-Slater system, constrained minimizer, blow-up behavior