Isometry-invariant solutions to a critical problem on non-compact Riemannian manifolds

We analyse an elliptic equation with critical growth set on a d-dimensional (d≥3) Hadamard manifold (M,g). By adopting a variational perspective, we prove the existence of non-zero non-negative solutions invariant under the action of a specific family of isometries. Our result remains valid when the original nonlinearity is singularly perturbed. Preserving the same variational approach, but considering other groups of isometries, we finally show that when M=Rd, d>3, and the nonlinearity is odd, there exist at least [Formula presented] pairs of sign-changing solutions.