Semilinear elliptic equations of the Hénon-type in hyperbolic space

Abstract

This paper deals with a class of the semilinear elliptic equations of the Hénon-type in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in [Formula: see text] Combining this compactness embedding with the Mountain Pass Theorem, a result of the existence of positive solution is established.

Infinitely many solutions for a Hénon-type system in hyperbolic space

“…for α, β ∈ R, N > 4 and obtained infinitely many non-trivial radial solutions. We would like to mention the paper of Carrião, Faria, and Miyagaki [4] where they extended He's result by considering a general nonlinearity…”

“…Problem (H * ) is closely related to the one studied by Carrião, Faria, and Miyagaki [4]. In [4], they proved that the map…”

“…Finally, we point out that since H is a closed subspace of the Hilbert space H 1 (B N ) × H 1 (B N ), following some ideas in [4,15], we can conclude that (u, v) is a critical point in…”

A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space $${{\mathbb {H}}}^N$$ H N

“…In particular, we would like to mention the Poincaré-Sobolev inequality of Mancini and Sandeep ([19], (1.2), which, as they have shown, by writing it in the half-space coordinates, follows from the Sobolev-Hardy-Mazy'a inequality, which is in turn equivalent to a subset of the original CKN inequalities by means of the ground state transform, also known as Picone identity); as well as related inequalities in [8] and [9]. Inequalities with weight play an important role in the study of Hénon-type equations in hyperbolic space, and a few such embeddings have been developed in [11] and [16]. The scale-invariant inequalities that we prove are significantly sharper than some of those found in literature.…”

“…The scale-invariant inequalities that we prove are significantly sharper than some of those found in literature. In particular, (30) is stronger than (1.1) in [19], while the weight in the embedding in [11], Lemma 2, case α = 0, which uses hyperbolic distance from the origin d(x) = log 1+r 1−r , behaves as a positive power of r at the origin and as a negative power of | log(1 − r )| at r = 1, while the weight (12) in our embeddings (31) and (26) has the power singularity both at the origin and at r = 1, see (19)- (20).…”