Positive singular solutions of a nonlinear Maxwell equation arising in mesoscopic electromagnetism

Abstract

At least one positive radial singular solution and infinitely many positive non-radial singular solutions of a nonlinear Maxwell equation in dimension two are constructed in a refined local version. The nonlinear Maxwell equation arises in mesoscopic electromagnetism. To construct the singular solutions, we need to know more detailed asymptotic behavior at the isolated singular point of the prescribed singular solutions than those already known. We will establish the asymptotic expansion up to arbitrary orders at the isolated singular point of a prescribed singular solution of the nonlinear Maxwell equation. Using such expansions, we construct positive singular solutions via fixed point arguments in some weighted Hölder spaces.