The elastic constants, electronic structure and optical properties of solid krypton are investigated by first-principles plane-wave pseudo-potential method in the scheme of density functional theory within local density approximation and generalized gradient approximation. The calculated elastic constants under pressure are in good agreement with the experimental and theoretical data. The obtained Debye temperature and sound velocity increase with pressure. The electronic structure and optical properties under pressure are also predicted, the energy band gap decreases, optical absorption coefficient increases and absorption peak widens under pressure. This leads to easy electron transition and solid krypton probably becomes semiconductor.