We present a new direction-of-arrival (DOA) algorithm which is applicable to the steering vector mismatch scenario based on sparse representation. In order to deal with the heavy-tailed noise in a real environment, we adopt the complex circularly symmetric generalized Gaussian distribution to simulate noise distribution. Given that the array steering vector can be transformed by sensor self motion and environmental factors, we employ the weighting least squares approach (WLS) to estimate the optimal gain value caused by  steering vector mismatch. Then we utilize sparse representation to reconstruct the signal model and convert the DOA estimation problem to a second-order cone programming (SOCP) problem. In order to reduce computational complexity, we adopt the singular value decomposition (SVD) method. Simulation results indicate that the proposed method can not only get high-resolution DOA estimation value, but  also effectively avoid the impact of steering vector mismatch on DOA estimation.