Large constant-sign solutions for boundray value problems (Bvps) of the partial discrete difference equation involving ( p,q)-Laplacian is investigated in this paper. The method is based on the critical point theory and truncation techniques. Even reducing to the ordinary difference equations, our results generalize and significantly improve some existing ones. In the end, the obtained results are exemplified based on two examples.

In this paper, we propose a dynamical model of AIDS which considers consciousness classification effect on vertical transmission. We separate the adult HIV infectives into two subclasses: ‘the unconscious class’ and ‘the conscious class’. We divide children into susceptible persons and AIDS patients. Furthermore, we consider the influence of susceptible persons and different infected persons on vertical transmission from mother to child. We calculate the basic reproduction number of the mathematical model. We explore the existence and local stability of all equilibria. The analysis further indicates that enhancing the HIV infectives’ awareness of disease and improving the proportion of conscious infectives could effectively control AIDS. Consequently, strengthening the publicity and education is of vital importance. We should improve the mass awareness and capability of disease prevention, thereby effectively reducing the contact rate. Moreover, we are supposed to enhance the HIV infectives’ consciousness of disease and improve the proportion of conscious infectives. Finally, a detailed conclusion ends the paper.