On the Absence of Positive Solutions of Elliptic Equations in Plane Unbounded Domains

Using model equations of the form $\Delta u+u^\sigma=0$ as an example, in both linear ($\sigma=1$) and nonlinear ($\sigma>1$) cases, we show some direct methods of proof of the absence of positive solutions in semi-infinite plane domains. We consider semi-infinite strips, both straight and exponentially decreasing, with homogeneous Neumann condition on the upper and lower boundaries.