Limiting behavior of minimizers of p(x)-Ginzburg-Landau type

Abstract

This paper is concerned with the asymptotic behavior of the radial minimizers of the p(x)-Ginzburg-Landau type functional. We prove the  uniqueness of radial minimizers in the case of 1 < p(x) < 2. In addition, this unique minimizer can be viewed as a limit of radial minimizers of a regularized functional. Based on these results, we obtain the Holder convergence by establishing the local W1;l-estimate. A new technique of counteracting the singularity plays a key role by estimating an accurate asymptotic rate. We believe that such a uniform estimate can provide some enlightenments how to handle other Ginzburg-Landau type equations, such as the p(x)-Laplace system without the radial structure.

Key words: p(x)-Ginzburg-Landau functional, uniqueness, radial minimizer, regularized functional, Holder convergence.