论文标题
摩尔斯索引与径向对称性的分数dirichlet问题
Morse index versus radial symmetry for fractional Dirichlet problems
论文作者
论文摘要
在这项工作中，我们提供了径向对称符号的摩尔斯索引，将有限的弱解决方案$ u $ $ u $ $ u $ $ $ $ $ $ $ $ $ $ $（-δ）^su = f（u）\ qquad \ qquad \ text {in $ \ nathcal {b} $} $ \ quad \ mathbb {r}^{n} \ setMinus \ mathcal {b} $，} $$其中$ s \ in（0,1）$，$ \ MATHCAL {b} \ subset \ subset \ subset \ subset \ mathbb {r}^n $是零和$ f $ f $ f $ f $ f $ f $ f $ f。我们证明，对于$ s \ in（1/2,1）$，上述问题的任何径向对称的符号更改解决方案都具有大于或等于$ n+1 $的莫尔斯索引。如果$ s \在（0,1/2]中，则$相同的结论在$ f $上的其他假设下。 $ \ Mathcal {B} $是反对称的。
In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-Δ)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet eigenvalue problem for the operator $(-Δ)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Bañuelos and Kulczycki.
