  We construct a singular solution of a stationary nonlinear Schr\"{o}dinger
equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear
behavior around zero. In view of Trudinger-Moser inequality, this type of
nonlinearity has an energy-critical growth. We use this singular solution to
prove non-uniqueness of strong solutions for the Cauchy problem of the
corresponding semilinear heat equation. The proof relies on explicit
computation showing a regularizing effect of the heat equation in an
appropriate functional space.
