We consider a semilinear elliptic equation on a smooth bounded domain $\Omega$ in $\mathbb R^2$, assuming that both the domain and the equation are invariant under reflections about one of the  coordinate axes, say the $y$-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the  axis, and, if  strictly positive, they are also decreasing in $x$ for $x>0$. Our goal is to exhibit examples of equations which admit nonnegative, nonzero solutions for which the second property fails; necessarily, such solutions have a nontrivial nodal set in $\Omega$. Previously, such examples were known for nonsmooth domains only.