  We construct a new class of positive solutions for a classical semilinear
elliptic problem in the plane which arise for instance as the standing-wave
problem for the standard nonlinear Schr\"odinger equation or in nonlinear
models in Turing's theory biological theory of pattern formation such as the
Gray-Scott or Gierer-Meinhardt systems. The solutions we construct have the
property that their energy over a ball of radius R grows linearly with R as R
tends to infinity. These solutions are strongly related to the solutions of a
Toda system.
