Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

The paper is concerned with positive solutions to problems of the type \[ -Δ_{BN} u - λ u = a(x) |u|ᵖ⁻¹\;u + f in BN, u ∈ H¹{(BN)}, \] where BN denotes the hyperbolic space, 1< p<2^*-1:=N+2/N-2 , \;λ < (N-1)²/4 , and f ∈ H⁻¹(BN) ( f ≡ 0 ) is a non-negative functional. The potential a∈ L^∞ (BN) is assumed to be strictly positive, such that lim d(x, 0) → ∞ a(x) → 1, where $d(x,\, 0)$ denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that a(x) ≤ 1 . Then the case a(x) ≥ 1 is considered, and the existence of two positive solutions is proved. In both cases, it is assumed that μ ( \ x : a(x) ≠ 1\) > 0. Subsequently, we establish the existence of two positive solutions for a(x) ≡ 1 and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles .