This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

中文翻译：

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平面对称和莫尔斯指数约束下的单调性结果

本文处理类型的半线性椭圆方程的解\[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{在 } \partial \Omega，\end{array} \right。\]其中 Ω 是平面的径向对称域，可以有界或无界。我们考虑解决方案你它们通过某个角度 θ 的旋转而不变，并且在这些旋转不变的函数空间中，它们的莫尔斯指数有界。我们可以证明或你是径向的，否则存在方向$e\in \mathcal {S}$这样你关于对称e并且在角度 θ/2 的扇区中的角度变量是严格单调的。该结果适用于旋转不变的函数空间中的最小能量和节点最小能量解，并产生多重结果。