Ground states for fractional magnetic operators

Abstract

Abstract. We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.

Bourgain–Brézis–Mironescu formula for magnetic operators

“…If A : R N → R N is a smooth function, the nonlocal operator dy, x ∈ R N , has been recently introduced in [6], where the ground state solutions of (−∆) s A u + u = |u| p−2 u in the three dimensional setting have been obtained via concentration compactness arguments. If A = 0, then the above operator is consistent with the usual notion of fractional Laplacian.…”

Fractional NLS equations with magnetic field, critical frequency and critical growth

“…In this paper, we study the following Schrödinger equations involving a critical nonlinearity u(x) − e i(x−y)·A( x+y 2 ) u(y) |x − y| N +2α dy, x ∈ R N , has been recently introduced in [13]. The motivations for its introduction are described in [13,32] in more detail and rely essentially on the Lévy-Khintchine formula for the generator of a general Lévy process.…”

“…The motivations for its introduction are described in [13,32] in more detail and rely essentially on the Lévy-Khintchine formula for the generator of a general Lévy process. If the magnetic field A ≡ 0, it seems that the first work which considered the existence of solutions for problem (1.1) in the subcritical case with ε = 1, formally α = 1 and K = 0 was [16].…”

“…For a wider treatment on these spaces, we refer the reader to [13]. Let L 2 (R N , C) be the Lebesgue space of complex-valued functions with summable square, endowed with the real scalar product…”

“…By virtue of the pointwise diamagnetic inequality [13,Remark 3.2] |u(x)| − |u(y)| ≤ u(x) − e i(x−y)·Aε( x+y 2 ) u(y) , for a.e. x, y ∈ R N and all ε > 0, we have…”