Nonlocal generalized models for a confined plasma in a Tokamak Original Research Article

The following model appears in plasma physics for a Tokamak configuration: −Δu + g(u) = 0, View the MathML sourceView the MathML source, View the MathML source, where I is a given positive constant, which is equivalent to find a fixed point u = F(u −g(u)) + ϕ0 where F is a compact operator on L2(Ω). According to Grad and Shafranov the nonlinearity g can depend on View the MathML source which is the generalized inverse of the distribution function m(t) = measx : u(x) > t = −vbu > t -vb (see [1]). But in these cases the map u → g(u) cannot be continuous on all the space View the MathML source but only on a nonlinear nonclosed set View the MathML source. This implies that the standard direct method for fixed point cannot be applied to solve the preceding problem. Nevertheless, using the Galerkin method and a topological argument, we prove that there exists a solution u fixed point of u = F(u − g(u)) + ϕ0 under suitable assumptions on g. The model we treat covers a large new class of nonlinearities including relative rearrangment and monotone rearrangment. The resolution of the concrete model needs an extension of the strong continuity result of the relative rearrangement map made in [2] (see Theorem 1.1 below for the definition and result).