Some existence results for the higher-order Brézis–Nirenberg problem on hyperbolic spaces

Abstract We study higher-order Brézis–Nirenberg type problems on hyperbolic spaces H n ${\\mathbb{H}}^{n}$ involving the 2 m -th order GJMS operator P m . First, for the Dirichlet problem P m u − λ u = | u | q − 2 u $${P}_{m}u-\\lambda u=\\vert u{\\vert }^{q-2}u$$ on bounded domains Ω ⊂ H n ${\\Omega}\\subset {\\mathbb{H}}^{n}$ with q = 2 n n − 2 m $q=\\frac{2n}{n-2m}$ and u ∈ W 0 m , 2 ( Ω ) $u\\in {W}_{0}^{m,2}\\left({\\Omega}\\right)$ , we establish existence and multiplicity of solutions when λ exceeds the first eigenvalue of P m , complementing the results of Li–Lu–Yang [J. Li, G. Lu, and Q. Yang, “Higher order Brezis–Nirenberg problem on hyperbolic spaces: existence, nonexistence and symmetry of solutions,” Adv. Math. , vol. 399, 2022, Art. no. 108259] for λ below the first eigenvalue. The proof relies on the symmetric mountain pass method, with energy estimates carried out via Helgason–Fourier analysis and truncation of the approximate Sobolev extremals on H n ${\\mathbb{H}}^{n}$ . Second, for the critical equation with nonlinear perturbation on the whole H n ${\\mathbb{H}}^{n}$ , P