论文标题
非线性Schrödinger方程的螺旋溶液
Spiraling solutions of nonlinear Schrödinger equations
论文作者
论文摘要
我们研究了固定非线性schrödinger方程$$-ΔV +q v = | v | v |^{p-2} v，\ qquad \ text {in $ \ mathbb {r}^3 $，带有$ 2 <p <p <\ iffty $和$ q \ ge 0 $ 0 $。这些溶液在某种意义上是螺旋式的，因为它们不是轴向对称的，而是在螺丝运动下不变的，即它们具有螺旋性的对称特性。除了存在结果外，我们还提供有关螺旋溶液形状的信息，该信息取决于代表基础螺钉运动的旋转斜率的参数值。我们的结果补充了Del Pino，Musso和Pacard在Allen-Cahn方程中的相关分析，而结果的性质和潜在的变分结构完全不同。
We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation $$ -Δv +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.
