Uniqueness of entire ground states for the fractional plasma problem

Authors: Hardy Chan, María Del Mar González (ICMAT-UAM), Yanghong Huang, Edoardo Mainini and Bruno Volzone

Source: Calculus of Variations and Partial Differential Equations

Date of publication: 26 October 2020



The equation  in dimension two has an interpretation in Plasma Physics since the spatial domain represents the cross section of a Totamak machine –a toroidal shell  containing a plasma ring surrounded by vacuum–. The region inhabited by the plasma is exactly the set {u>C}, with u modeling the flux function.

In this work we consider its factional version, in which the diffusion given by the Laplacian –which is a local operator–, is replaced by a non-local diffusion defined from an integro-differential operator. We prove uniqueness of ground states, which are entire, radially symmetric and decreasing solutions in the radial variable. The main difficulty comes from the non-locality of the equation since, contrary to the local case, it cannot be reduced to the study of an ODE. Instead, we use a monotonicity argument for PDEs.