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

Abstract

The equations under consideration have the following structure: $$\frac{{\partial ^2 u}}{{\partial x_n^2 }} + \sum\limits_{i,j = 1}^{n - 1} {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{{\partial u}}{{\partial x_j }}} \right)} + \sum\limits_{i = 1}^{n - 1} {a_i (x)\frac{{\partial u}}{{\partial x_i }}} - f(u,x_n ) = 0,$$ where 0 < xn < ∞, (x1, …, xn−1) ∈ Ω, Ω is a bounded Lipschitz domain, \(f(0,x_n ) \equiv 0,\tfrac{{\partial f}}{{\partial u}}(0,x_n ) \equiv 0\) 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 xn → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas were obtained in the case of aij, ai depending only on (x1, …, xn−1).