On the asymptotic behavior of solutions to a semilinear elliptic boundary problem in an unbounded domain

We consider solutions to the elliptic linear equation (1) of second order i n an unbounded domain Q in R-n supposing that Q = {x = (x', x(,)) :0 < x(n) < infinity, /x'/ < gamma(x(n))}, where 1 less than or equal to gamma(t) le ss than or equal to At + B, and that u satisfy the boundary condition (2). We show that any such solution u growing moderately at infinity is bounded and tending to 0 as x(n) --> infinity. Earlier we showed in our notes [3,4] this theorem for the case gamma(x(n)) = B, i.e., for a cylindrical domain Q = Omega x (0, infinity), Omega subset of Rn-1. (C) 2000 Academic des scie nces/Editions scientifiques et medicales Elsevier SAS.