ASYMPTOTIC-BEHAVIOR OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC-EQUATIONS ON EXPANDING ANNULI

We study the asymptotic behavior of positive solutions of the semilinear elliptic equation Delta u + f(u) = 0 in Omega(a), u = 0 on partial derivative Omega(a), where Omega(a),= {x is an element of R(N): a < \x\ < a + 1} are expanding annuli as a --> infinity, and Sis positive and superlinear at both 0 and infinity. We first show that there are a priori bounds for some positive solutions u(a)(x) as a --> infinity. Then, if we fu: any direction, after a suitable translation of u(a) the limiting solutions are non-negative solutions on the infinite strip. We can obtain more detailed descriptions of these limits if u(a) is radially symmetric, least-energy, or least-energy with a particular symmetry. (C) 1995 Academic Press, Inc.