Abstract: The solutions to semilinear elliptic partial differential equations contain rich information about the equations, which is very important for describing the development of various phenomena. The existence of equilibrium solutions of multi-species mutual aid model and the economic equilibrium point can be transformed into the existence of the solutions to Neumann boundary value problems. In this paper, we study the existence and uniqueness of the solutions for a class of semilinear elliptic equations with Neumann boundary value conditions. Using the topological degree theory and the eigenvalue comparison principle, we obtain the existence of the solutions under the assumption that the nonlinear terms satisfy the asymptotic nonuniform conditions. Using the eigenvalue comparison principle, we prove the uniqueness of the solutions. The obtained results extend and complement some relevant existing works. As an application, an example is given to verify the obtained results.

Key words: Neumann boundary value problem, existence and uniqueness of solutions, asymptotic nonuniform condition, topological degree, the eigenvalue comparison principle