Maximum principles for inhomogeneous equation

A class of inhomogeneous semilinear elliptic equations is considered. The Hopf maximum principles are used to deduce that certain functions defined for solutions of the equation attain a maximum on the boundary of the domain or at a critical point of the solution. Dirichlet problem and Neumann problem are considered also. These maximum principles may be used to determine bounds for quantities of interest in some physical problems. Our results generalize and deepen those from corresponding work in [5, 6].