Symmetry of positive solutions to semilinear elliptic problems in half space (Q1599976)

This paper deals with the following conjecture of H. Berestycki: Let \(u\) be positive bounded solution of \[ \begin{cases} \Delta u+f(u)= 0\quad & \text{in }H= \{(x,y)\in\mathbb{R}^N\mid x> 0\},\\ u= 0\quad &\text{on }\partial H,\end{cases}\tag{1} \] and let \(M= \text{sup }u\). If there is a bounded solution of (1), then necessarily \(f(M)= 0\). In this note, using various forms of maximum principles and the method of moving planes, the author proves the conjecture.