A Note on Static Solutions of a Lorentz Invariant Equation in Dimension 3

The aim of this Letter is to prove the existence of a static solution to the Lorentz invariant equation □2u+ε□6u+V′(u)=0 in every class of maps with nonzero topological charge when the singular potential V has some radial symmetry. Here u:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}3+1→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}4, u=u(x,t), xε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document}3, tε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\kern 1pt} \square _{{\kern 1pt} p^u } = \frac{\partial }{{\partial _t }}\left[ {\left( {c^2 \left| {\nabla u} \right|^2 - \left| {u_t } \right|^2 } \right)^{p - 2} } \right] - c^2 \nabla \left[ {\left( {c^2 \left| {\nabla u} \right|^2 - \left| {u_t } \right|^2 } \right)^{p - 2} \nabla u} \right].$$ \end{document}