Existence of High Energy-Positive Solutions for a Class of Elliptic Equations in the Hyperbolic Space.

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space: - Δ B N u - λ u = a (x) | u | p - 1 u in B N , u ∈ H 1 (B N) , where B N denotes the hyperbolic space, 1 < p < 2 ∗ - 1 : = N + 2 N - 2 , if N ⩾ 3 ; 1 < p < + ∞ , if N = 2 , λ < (N - 1) 2 4 , and 0 < a ∈ L ∞ (B N). We prove the existence of a positive solution by introducing the min–max procedure in the spirit of Bahri–Li in the hyperbolic space and using a series of new estimates involving interacting hyperbolic bubbles. [ABSTRACT FROM AUTHOR]