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 appears 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 Willem (Minimax theorem. Progress in nonlinear differential equations and their applications, vol 24, Birkhäuser Boston Inc., Boston, 1996). 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 noncompact type.

例如，对称性在变分问题（已解决，例如\（{\ mathbb {R}} ^ {n} \或更一般的流形）中起着基本作用，以解决自然而然出现的紧凑性不足当问题在非紧缩组的作用下不变时。在\（{\ mathbb {R}} ^ n \）中，已经证明了在G动作的条件下，相对于\（\ mathrm {O}（n）\）的子组G的不变函数的紧致结果。在\（{\ mathbb {R}} ^ n \）上关于相容性，请参见Willem（Minimax定理。非线性微分方程及其应用的进展，第24卷，BirkhäuserBoston Inc.，波士顿，1996年）。作为第一个结果，我们对此进行了概括，并在此处表明，对于黎曼流形的等轴测图组的特定子组，恢复了紧实度。我们还研究了对Hadamard流形（M，g）的等距作用， 证明\（\ mathrm {Iso}（M，g）\）的一大类子集是兼容的。作为应用，我们得到了“不变”功能的紧凑性的结果，使我们能够证明非径向解的存在性的经典标量方程和在外地分数方程\（{\ mathbb {R}} ^ N \）为\（n = 3 \）和\（n = 5 \），改善了文献中已知的一些结果。最后，我们证明了非径向不变函数的存在，使得对于非紧型的某些对称空间，紧致性结果成立。