论文标题
紧凑性引理的理论证明和半线性椭圆方程的非自由解的存在
A group theoretic proof of a compactness lemma and existence of nonradial solutions for semilinear elliptic equations
论文作者
论文摘要
对称性在变异问题中起着基本作用（例如在$ \ mathbb r^{n} $中或更一般的歧管中定居），例如，在不相处组的动作下，当问题是不变时出现的缺乏紧凑性。在$ \ mathbb r^n $中，在$ \ \ m athbb r^n $上的$ g $ Action是在$ g $ action的条件下证明了不变函数的紧凑性结果。作为第一个结果，我们将其概括并在这里表明，对于riemannian歧管的等轴测组的特定亚组，恢复了紧凑性。我们还研究了Hadamard歧管$（m，g）$的等距操作，证明了$ \ mathrm {iso}（m，g）$的一大类子组是兼容的。作为应用程序，我们获得了``不变式''函数的紧凑性结果，该功能使我们能够证明存在经典标量方程的非自由解决方案，以及$ \ mathbb r^n $ for $ n = 3 $ = 3 $ n = 3 $ n = 3 $和$ n = 5 $的非局部分数方程，改善了文献中已知的一些结果。最后，我们证明了非放射不变函数的存在，因此对于某些非紧凑型类型的对称空间，紧凑性结果具有。
Symmetry plays a basic role in variational problems (settled e.g. in $\mathbb R^{n}$ or in a more general manifold), for example to deal with the lack of compactness which naturally appear when the problem is invariant under the action of a noncompact group. In $\mathbb R^n$, a compactness result for invariant functions with respect to a subgroup $G$ of $\mathrm{O}(n)$ has been proved under the condition that the $G$ action on $\mathbb R^n$ is compatible, see \cite{willem}. As a first result we generalize this and show here that the compactness is recovered for particular subgroups of the isometry group of a Riemannian manifold. We investigate also isometric action on Hadamard manifold $(M,g)$ proving that a large class of subgroups of $\mathrm{Iso}(M,g)$ is compatible. As an application we get a compactness result for ``invariant'' functions which allows us to prove the existence of nonradial solutions for a classical scalar equation and for a nonlocal fractional equation on $\mathbb R^n$ for $n=3$ and $n=5$, improving some results known in the literature. Finally, we prove the existence of nonradial invariant functions such that a compactness result holds for some symmetric spaces of non compact type.
