Galerkin approximation, strong continuity of the relative rearrangement map and application to plasma physics equations (Q1854058)

The Galerkin method is applied to solve the nonlinear equation \(\Delta u = F(u)\) in an open bounded connected set of \(\mathbb{R}^N\), \(N\geq 2\). Based on the Brouwer fixed-point theorem, Sobolev space techniques and eigenfunctions of Laplace operator with Dirichlet boundary conditions, the author proves the unique solvability of this problem. The theory is applied to plasma physics equations.