In low-earth orbit (LEO) satellite communications, it is difficult to obtain accurate channel estimation results in the classical time-frequency (TF) domain due to the large Doppler shift. Orthogonal time frequency space (OTFS) modulation is a potential solution since it can convert the time-variant channels in the TF domain into the time-invariant channels in the delay-Doppler (DD) domain. In this paper, we focus on multi-user random access in the LEO satellite communication system, where multiple users with identical pilot sequences will significantly reduce the user detection rate and CE accuracy. Therefore, we propose an OTFS pilot design based on the delay and Doppler spread to ensure the orthogonality of the received pilots. Additionally, we propose a channel estimation (CE) algorithm based on the sparsity of the channel in the delay-Doppler-angular domain to improve multi-user separation accuracy. Simulation results illustrate that the proposed algorithm can effectively distinguish multiple users with identical pilot sequences and improve channel estimation accuracy.