Spiraling solutions of nonlinear Schr\"odinger equations

We study a new family of sign-changing solutions to the stationary nonlinear Schr\"odinger equation $$-\Delta v +q v =|v|^{p-2} v, \qquad \text{in $\mathbb{R}^3$,}$$ with $2<p<\infty$ and $q \ge 0$. These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard for the Allen-Cahn equation, whereas the nature of results and the underlying variational structure are completely different.