Asymptotic properties of solutions of semilinear second-order elliptic equations in cylindrical domains

The equations under consideration have the following structure: ∂2u/∂x2n + n-1 ∑ i,j=1 ∂/∂xi(aij(x)∂u/∂xj) + n-1 ∑ i=1 ai(x)∂u/∂xi - f(u,xn)=0, where 0 < x n < ∞, (x 1,⋯, x n-1) Ω, Ω is a bounded Lipschitz domain, f(0,xn) ≡ 0, ∂f/∂u(0, xn ≡ 0, f is a function that is continuous and monotonic with respect to u, and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations as x n → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas were obtained in the case of a ij, ai depending only on (x 1,⋯, x n-1). © 2006 Springer Science+Business Media, Inc.