Abstract: We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space
−ΔBNu−λu=a(x)up−1+εu2∗−1in BN,u∈H1(BN),
where BN denotes the hyperbolic space, 2<p<2∗:=2NN−2, if N⩾3;2<p<+∞, if N=2, λ<(N−1)24, and 0<a∈L∞(BN). We first prove the existence of a positive radially symmetric ground-state solution for a(x)≡1. Next, we prove that for a(x)⩾1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in RN. Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a(x)⩽1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.
Keywords: Hyperbolic space, hyperbolic bubbles, Palais–Smale decomposition, semilinear elliptic problem
