Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space

Abstract In this article, we study the following stationary Schrödinger equation on hyperbolic space: − Δ H N u + λ u = f ( u ) , x ∈ H N , N ≥ 3 , -{\Delta }_{{{\mathbb{H}}}^{N}}u+\lambda u=f\left(u),\hspace{1.0em}x\in {{\mathbb{H}}}^{N},\hspace{1em}N\ge 3, where Δ H N {\Delta }_{{{\mathbb{H}}}^{N}} denotes the Laplace-Beltrami operator on H N {{\mathbb{H}}}^{N} , λ ∈ R \lambda \in {\mathbb{R}} , and f f is locally Lipschitz continuous satisfying asymptotically linear at infinity. After a reduction from hyperbolic case to Euclidean case, using variational methods, we prove the existence of positive solutions for the aforementioned equation under suitable conditions on λ \lambda and f f .