Laguerre-Gaussian (LG) vortex beams are one of the current research hotspots, and with the further research, their transmission characteristics in the medium have also been paid attention to. Based on the nonlinear wave equation, the split-step Fourier algorithm is used to calculate the amplification characteristics of LG vortex beam in Kerr medium. The effects of topological charge, beam waist width and incident beam peak intensity on the intensity, spectrum, orbital angular momentum (OAM) spectrum and phase distribution of LG vortex beams are investigated. The numerical results show that after the LG vortex beam is amplified by Kerr medium, the beam distribution gradually evolves from circular to ellipsoidal, and the radius of the central sag decreases gradually. The light intensity is enhanced, and the light intensity in the enhanced region fluctuates. At different azimuths, the peak light intensity will shift inward (θ=π/4) or outward (θ=π/2). The spectrum intensity is enhanced and the width is widened. In addition, the beam that produces the new OAM component will be stimulated. The influence of Kerr effect on beam amplification can be reduced by decreasing topological charge, incident peak light intensity and increasing beam waist width, and the B-integral can be reduced effectively.