We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger 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 in their study (2012, Manuscripta Math., 138, 273–286) for the Allen–Cahn equation, whereas the nature of results and the underlying variational structure are completely different.

我们研究了平稳非线性薛定谔方程 \[ -\Delta v +qv =|v|^{p-2} v, \qquad \text{in}\,{ {\mathbb{ R}^{3}},} \] 与$2 < p < \infty$和$q \ge 0$。这些解在某种意义上是螺旋形的，它们不是轴对称的，而是在螺旋运动下不变的，即它们共享螺旋面的对称特性。除了存在结果之外，我们还提供了关于螺旋解的形状的信息，这取决于表示底层螺旋运动的旋转斜率的参数值。我们的结果补充了 Del Pino、Musso 和 Pacard 在他们的研究（2012，Manuscripta Math., 138, 273-286) 对于 Allen-Cahn 方程，而结果的性质和潜在的变分结构完全不同。