Positive solutions to a class of second-order semilinear elliptic equations in an exterior domain (Q1935006)

In this article, the authors study the following semilinear elliptic equation: \[ \leqno(1)\quad\quad \Delta u+f(x, u, \nabla u)=0, \quad x\in \Omega_A, \] where \(\Omega_A=\{x\in {I\!\!R}^n, |x|>A\}\) is an exterior domain of \({I\!\!R}^n,\) with \(n\geq 3, A\geq 1\) and \(f\) is a locally Hölder continuous function in \(\Omega_A\times {I\!\!R} \times{I\!\!R}^n.\) The authors combine the sub- and supersolution method and the Schauder-Tikhonov fixed point theorem to prove the existence of a positive solution of the problem (1) with \(\displaystyle\lim_{|x|\rightarrow \infty} u(x)=0.\)