Liouville Theorems for critical points of the p-Ginzburg-Landau type functional

TL;DR

This paper establishes Liouville theorems for critical points of the p-Ginzburg-Landau functional on Riemannian manifolds, showing under certain curvature and growth conditions that solutions are trivial or constant.

Contribution

It introduces new Liouville theorems for p-Ginzburg-Landau critical points under curvature and growth assumptions, extending previous results to more general settings.

Findings

Liouville theorems for p-Ginzburg-Landau critical points on curved manifolds

Uniqueness of constant solutions in Dirichlet problems on starlike domains

Conditions under which solutions must be trivial or constant

Abstract

In this paper, we consider the smooth map from a Riemannian manifold to the standard Euclidean space and the p-Ginzburg-Landau energy. Under suitable curvature conditions on the domain manifold, some Liouville type theorems are established by assuming either growth conditions of the p-Ginzburg-Landau energy or an asymptotic condition at the infinity for the maps. In the end of paper, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains.