论文标题
在混合型操作员的光谱上，应用于旋转波溶液
On the spectrum of a mixed-type operator with applications to rotating wave solutions
论文作者
论文摘要
我们研究非线性波方程的旋转波解$$ \ weft \ {\ oken {Aligned} \ partial_ {t}^2 v -ΔV + m v＆v＆v＆v＆v＆v＆v | v |^{p -2} v \ quad && quad && pext { \ text {on $ \ mathbb {r} \ times \ partial \ mathbf {b} $} \ end end {aligned} \ right。 $$ $ 2 <p <p <\ infty $，$ m \ in \ mathbb {r} $和$ \ mathbf {b} \ subset \ mathbb {r}^2 $表示单位磁盘。如果旋转的角速度$α$大于$ 1 $，则会导致涉及混合型操作员的$ \ mathbf {b} $上的半线性边界价值问题，其光谱与贝塞尔功能的零有关，并且通常行为不佳。基于这些零的新估计值，我们发现$α$的值，因此该频谱仅由具有有限多重性的特征值组成，并且没有累积点。结合适当的光谱估计，这使我们能够制定适当的无限变化设置，并找到$ p \ in（2,4）$的降低方程的基态解决方案。使用基态能量的最小值表征，我们最终表明这些接地状态是非自由的，因此产生了非平凡的旋转波，前提是$ m $足够大。
We study rotating wave solutions of the nonlinear wave equation $$ \left\{ \begin{aligned} \partial_{t}^2 v - Δv + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \mathbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \mathbf{B}$} \end{aligned} \right. $$ where $2<p<\infty$, $m \in \mathbb{R}$ and $\mathbf{B} \subset \mathbb{R}^2$ denotes the unit disk. If the angular velocity $α$ of the rotation is larger than $1$, this leads to a semilinear boundary value problem on $\mathbf{B}$ involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of $α$ such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for $p \in (2,4)$. Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided $m$ is sufficiently large.
