We consider the problem $\Delta u+V(x)u=f'(u)$ in $\mathbb R^N$. Here the nonlinearity has a double power behavior and $V$ is invariant under an orthogonal involution, with $V(\infty)=0$. An existence theorem of one pair of solutions which changes sign exactly once is given.
"Existence of minimal nodal solutions for the nonlinear Schrödinger equations with $V(\infty)=0$." Adv. Differential Equations 11 (12) 1375 - 1396, 2006.