A mathematical form of force-free magnetosphere equation around Kerr black holes and its application to Meissner effect

Based on the Lagrangian of the steady axisymmetric force-free magnetosphere (FFM) equation around Kerr black holes(KBHs), we find that the FFM equation can be rewritten in a new form as $f_{,rr} / (1-\mu^{2}) + f_{,\mu\mu} / \Delta + K(f(r,\mu),r,\mu) = 0$,where$\mu = -\cos\theta$. By coordinate transformation, the form of the above equation can be given by $s_{,yy} + s_{,zz} + D(s(y,z),y,z) = 0$. Based on the form, we prove finally that the Meissner effect is not possessed by a KBH-FFM with the condition where $d\omega \mathrm{/}d{A}_{\varphi }⩽0$ and $H_{\phi}(dH_{\phi}/dA_{\phi}) \geqslant 0$,here$A_{\phi}$isthe$\phi$componentofthevectorpotential$\vec{A}$,$\omega$istheangularvelocityofmagneticfieldsand${H_{\phi}}$ corresponds to twice the poloidal electric current.