The variable-coefficient partial differential equations are not only used in many physical models, but also fundamentally applied in the field of nonlinear science. In order to solve certain variable-coefficient partial differential equations, the auxiliary elliptic-like equation method is introduced in this article by means of the symbolic computation software. The basic idea of the new algorithm is that if certain variable-coefficient partial differential equation can be converted into the form of elliptic equation, then its solutions are readily obtained. By taking the Kadomtsev-Petviashvili equation for an example, not only the effectiveness of the algorithm is demonstrated, but many new solutions are worked out, including dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions, which may be useful for depicting  nonlinear physical phenomena.