Uniqueness of non-linear ground states for fractional Laplacians in ${\mathbb{R}}$

An extension problem for the CR fractional Laplacian

“…This idea of using the extension has been successfully employed in several other settings ( [27,28,4], for instance).…”

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

“…When γ = 0 and α ∈ (0, 1), the problem (1.1) is a nonlocal model known as nonlinear fractional Schrödinger equation which has also attracted much attentions recently [9,10,11,12,19,20,21,22,23,24,15,16]. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was derived by Laskin [29,30] as a result of extending the Feynman path integral, from the Brownian-like to Levy-like quantum mechanical paths.…”

Existence and mass concentration of pseudo-relativistic Hartree equation

“…It is known from [8,16] that, up to translations, problem (1.8) admits a positive ground state Q, which can be taken to be radially symmetric about the origin. Moreover, every ground state Q of (1.8) satisfies Q ∈ H s (R 3 ) for all s ≥ 1 2 and…”

“…The optimal constant in inequality (1.11) was determined in [8]. Taking into account as [20] the Weinstein functional 12) we know from Lemma 5 in [18] that 13) and then, the best constant C gn in (1.11) is equal to 2 Q 2 2 .…”