We apply the trajectory dynamical systems approach to study the nonnegative solutions of a semilinear elliptic problem in an unbounded domains. The existence of the global attractor for the trajectory dynamical system associated with this problem is proved. The symmetrization and stabilization properties of nonnegative solutions as |x| ∞ are also established under new (weak) conditions in three dimensional case Ω ⊂ ℝ³