The existence of a (unique) solution of the second-order semilinear elliptic equation $$\sum\limits_{i,j = 0}^n {a_{ij} (x)u_{x_i x_j } + f(\nabla u,u,x) = 0}$$ withx=(x0,x1,⋯,xn)∃(s0, ∞)× Ω′, for a bounded domainΩ′, together with the additional conditions $$\begin{array}{*{20}c} {u(x) = 0for(x_1 ,x_2 ,...,x_n ) \in \partial \Omega '} {u(x) = \varphi (x_1 ,x_2 ,...,x_n )forx_0 = s_0 } {|u(x)|globallybounded} \end{array}$$ is shown to be a well-posed problem under some sign and growth restrictions off and its partial derivatives. It can be seen as an initial value problem, with initial valueϕ, in the spaceC00\((\overline {\Omega '} )\) and satisfying the strong order-preserving property. In the case thataij andf do not depend onx0 or are periodic inx0, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions onf are given under which all the solutions tend to zero asx0 tends to infinity. Proofs are strongly based on maximum and comparison techniques.