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Based on the Lagrangian of the steady axisymmetric force-free magnetosphere(FFM) equation around Kerr black holes(KBHs), we find that the FFM equation canbe 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 coordinatetransformation, 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 theMeissner effect is not possessed by a KBH-FFM with the condition where$d\omega/d A_{\phi} \leqslant 0$ and $H_{\phi}(dH_{\phi}/dA_{\phi}) \geqslant0$, here $A_{\phi}$ is the $\phi$ component of the vector potential $\vec{A}$,$\omega$ is the angular velocity of magnetic fields and ${H_{\phi}}$corresponds to twice the poloidal electric current.

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