Relativistic equations with singular potentials

Abstract

The first part of this paper concern with the study of the Lorentz force equation q′1-|q′|2′=E→(t,q)+q′×B→(t,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( \\frac{q'}{\\sqrt{1-|q'|^2}}\\right) '= \\overrightarrow{E}(t,q)+q'\ imes \\overrightarrow{B}(t,q) \\end{aligned}$$\\end{document}in the relevant physical configuration where the electric field overrightarrow{E} has a singularity in zero. By using Szulkin’s critical point theory, we prove the existence of T-periodic solutions provided that T and the electric and magnetic fields interact properly. In the last part, we employ both a variational and a topological argument to prove that the scalar relativistic pendulum-type equation q′1-(q′)2′+q=G′(q)+h(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( \\frac{q'}{\\sqrt{1-(q')^2}}\\right) ' +q = G^{\\prime }(q) +h(t), \\end{aligned}$$\\end{document}admits at least a periodic solution when hin L^1 (0, T) and G is singular at zero.