On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations

We study the Dirichlet problem in half-space for the equation $\nobreak{\Delta u+g(u)\vert\nabla u\vert^2=0,}$ where $g$ is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as $y\to\infty,$ where $y$ denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.