Relativistic equations with singular potentials

Abstract

AbstractThe first part of this paper concern with the study of the Lorentz force equation $$\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '= \overrightarrow{E}(t,q)+q'\times \overrightarrow{B}(t,q) \end{aligned}$$ q ′ 1 - | q ′ | 2 ′ = E → ( t , q ) + q ′ × B → ( t , q ) in the relevant physical configuration where the electric field $$\overrightarrow{E}$$ 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 $$\begin{aligned} \left( \frac{q'}{\sqrt{1-(q')^2}}\right) ' +q = G^{\prime }(q) +h(t), \end{aligned}$$ q ′ 1 - ( q ′ ) 2 ′ + q = G ′ ( q ) + h ( t ) , admits at least a periodic solution when $$h\in L^1 (0, T)$$ h ∈ L 1 ( 0 , T ) and G is singular at zero.