In this paper, we investigate the qualitative properties of positive solutions for a two-coupled elliptic system in the punctured space. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity.

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In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced ``transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.

Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the H\"older and the gradient $L^p$ estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions.

The compressible Euler equations, which govern the gas flow surrounding a solid ball with mass M and frictional damping in n dimensions, t+(u)= 0,(u) t+ (u u)+ P ()=-Mx/| x| n-2u, where , u, P and M are the density, velocity, pressure and mass of the gas, respectively, n 3 is the dimension of x, and > 0 is the frictional constant, are examined. The pressure is assumed to satisfy the law and 12 , K is a positive constant. The existence and non-existence of global smooth solutions are studied for the initial boundary problem of the Euler equations.