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Unsupervised domain adaptation~(UDA) aims at reducing the distribution discrepancy when transferring knowledge from a labeled source domain to an unlabeled target domain. Previous UDA methods assume that the source and target domains share an identical label space, which is unrealistic in practice since the label information of the target domain is agnostic. This paper focuses on a more realistic UDA scenario, i.e. partial domain adaptation (PDA), where the target label space is subsumed to the source label space. In the PDA scenario, the source outliers that are absent in the target domain may be wrongly matched to the target domain (technically named negative transfer), leading to performance degradation of UDA methods. This paper proposes a novel Target Domain Specific Classifier Learning-based Domain Adaptation (TSCDA) method. TSCDA presents a soft-weighed maximum mean discrepancy criterion to partially align feature distributions and alleviate negative transfer. Also, it learns a target-specific classifier for the target domain with pseudo-labels and multiple auxiliary classifiers, to further address classifier shift. A module named Peers Assisted Learning is used to minimize the prediction difference between multiple target-specific classifiers, which makes the classifiers more discriminant for the target domain. Extensive experiments conducted on three PDA benchmark datasets show that TSCDA outperforms other state-of-the-art methods with a large margin, e.g. $4\%$ and $5.6\%$ averagely on Office-31 and Office-Home, respectively.