Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space

Abstract

In this article, we study the following stationary Schrödinger equation on hyperbolic space: −ΔHNu+λu=f(u),x∈HN,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 ΔHN{\Delta }_{{{\mathbb{H}}}^{N}} denotes the Laplace-Beltrami operator on HN{{\mathbb{H}}}^{N}, λ∈R\lambda \in {\mathbb{R}}, and ff 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 ff.

Keywords

• asymptotically linear schrödinger equation
• hyperbolic space
• mountain pass theorem
• linking theorem
• 35a01
• 35a15
• 35b09