Bounded solutions of some nonlinear elliptic equations in cylindrical domains

The existence of a (unique) solution of the second-order semilinear elliptic equation ∑i.j = 0n aij(x) uxixj + f(∇u, u, x) = 0 with x = (x0, xl,..., xn) ∈ (s0, ∞) × Ω′, for a bounded domain Ω′, together with the additional conditions u(x) = 0 for (x1, x2,..., xn) ∈ ∂Ω′ u(x) = φ(x1, x2,..., xn) for x0 = s0 |u(x)| globally bounded is shown to be a well-posed problem under some sign and growth restrictions of f and its partial derivatives. It can be seen as an initial value problem, with initial value φ, in the space ℓ00(Ω̄′) and satisfying the strong order-preserving property. In the case that aij and f do not depend on x0 or are periodic in x0, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions on f are given under which all the solutions tend to zero as x0 tends to infinity. Proofs are strongly based on maximum and comparison techniques. © 1997 Plenum Publishing Corporation.