Abstract

We study the positive solutions of a semilinear elliptic problem in an asymptotically symmetric unbounded domain Ω + in ℝ 4 . The existence of the global attractor for the trajectory dynamical system associated with this problem is proved. Based on the recent development of the De Giorgi conjecture, the symmetrization and stabilization properties of positive solutions as | x |→∞ are also established in the four-dimensional case.