In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equa-tions in R3 . Let ωθ , Jθ and uθ be the azimuthal component of ω , J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution ( u, b ) is regular on (01, T ) if ( ωθ1, Jθ ) ∈ Lq (01, T ; Lp ) or ( ωθ , ( uθeθ )) ∈ Lq (01,T ; Lp ) with 3/p + 2/q ≤ 2, 3/2 < p < ∞ . In the endpoint case, one needs conditions ( ωθ1, Jθ ) ∈ L1 (01, T;B0∞,∞ ) or (ωθ1,▽(uθeθ )) ∈ L1 (01,T;B0∞,∞ ).
Using the fibering method introduced by Pohozaev, we prove existence of positive solution for a Diriclhlet problem with a quasilinear system involving p-Laplacian operator.
