Existence and blow-up behavior of constrained minimizers for Schrödinger-Poisson-Slater system

Existence and limiting behavior of minimizers for attractive Schrödinger‐Poisson systems with periodic potentials

“…Indeed, let 0 < ≤ 4 3 and m > 0. Let ∈ H 1 be such that || || 2 L 2 = m and denote as in (25). We see that || || 2 L 2 = a for any > 0 and…”

“… However, the appearance of periodic potentials makes the proof more involved. The first difficulty is that the energy functional is not well behaved under the mass scaling , which plays an important role in the previous studies . The second one is the lack of compactness at infinity.…”

“…Remark In the case of no periodic potential, Zhu proved the existence of minimizers for …”

“…In the case of attractive interaction = −1, the existence of standing waves and its limiting behavior has been considered in recent works. 24,25 In this paper, we study the existence, nonexistence, and limiting behavior of minimizers for the energy functional related to attractive Schrödinger-Poisson systems = −1 with periodic potentials. To our knowledge, this paper seems to be the first one dealing with the prescribed mass minimization of the energy functional related to the Schrödinger-Poisson system with periodic potentials.…”

“…The first difficulty is that the energy functional is not well behaved under the mass scaling (x) ∶= ( x), which plays an important role in the previous studies. 4,7,16,17,25 The second one is the lack of compactness at infinity. Note that in the case of trapping potentials, 23 the trapping condition lim |x|→∞ V(x) = ∞ ensures the Hilbert space…”