On fractional plasma problems

Abstract

In this paper we show existence and multiplicity results for a linearly perturbed elliptic problem driven by nonlocal operators, whose prototype is the fractional Laplacian. More precisely, when the perturbation parameter is close to one of the eigenvalues of the leading operator, the existence of three nontrivial solutions is proved.

Uniqueness of entire ground states for the fractional plasma problem

“…In particular the author in [1] studies existence and regularity of solutions, and the nonlocal counterpart of the geometry of the free boundary ∂ {u = C}, which was previously obtained in [33]. Recently, in [15] some interesting existence results are established by critical point theory for the eigenvalue problem related to a general nonlocal operator L K with a singular kernel K (note that L K = (− ) s for the choice K (x) = |x| −N −2s ), i.e. the problem…”

Uniqueness of entire ground states for the fractional plasma problem

“…In particular the author in [1] studies existence and regularity of solutions, and the nonlocal counterpart of the geometry of the free boundary ∂ {u = C}, which was previously obtained in [31]. Recently, in [14] some interesting existence results are established by critical point theory for the eigenvalue problem related to a general nonlocal operator L K with a singular kernel K (note that L K = (−∆) s for the choice K(x) = |x| −N −2s ), i.e. the problem…”

Existence of infinitely many high energy solutions for a fractional Schrödinger equation in RN

“…The fractional Schrödinger equation was introduced by Laskin [14,15] in the context of fractional quantum mechanics, as a result of extending the Feynman path integral from the Brownian-like to Lévy-like quantum mechanical paths. It is also appeared in several subjects such as plasma physics, image processing, finance and stochastic models, see for instance [1,4,10,16].…”