Infinitely many solutions for a H\'enon-type system in hyperbolic space

This paper is devoted to study the semilinear elliptic system of H\'enon-type \begin{eqnarray*} -\Delta_{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v) \\ -\Delta_{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v) \end{eqnarray*} in the hyperbolic space $\mathbb{B}^{N}$, $N\geq 3$, where $u, v \in H_{r}^{1}(\mathbb{B}^{N})=\{\phi\in H^1(\mathbb{B}^N): \phi\, \text{is radial}\}$ and $-\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^N$, $Q \in C^{1}(\mathbb{R}\times \mathbb{R},\mathbb{R})$ is a p-homogeneous function, $d(x)=d_{\mathbb{B}^N}(0,x)$ and $K\geq0 $ is a continuous function. We prove a compactness result and together with the Clark's theorem we establish the existence of infinitely many solutions.


Introduction and main result
This article concerns the existence of infinitely many solutions for the following semilinear elliptic system of Hénon type in hyperbolic space where B N is the Poincaré ball model for the hyperbolic space, H 1 r (B N ) denotes the sobolev space of radial H 1 (B N ) function, r = d(x) = d B N (0, x), ∆ B N is the Laplace-Beltrami type operator on B N .
We assume the following hypothesis on K and Q (K 1 ) K ≥ 0 is a continuous function with K(0) = 0 and K = 0 in B N \{0}.
In the past few years the prototype problem has been attracted attention. Unlike the corresponding problem in the Euclidean space R N , He in [10] proved the existence of a positive solution to the above problem over the range p ∈ (2, 2N +2α N −2 ) in the hyperbolic space. More precisely, she explored the Strauss radial estimate for hyperbolic space together with the Mountain Pass Theorem. In a subsequent paper [11], she proved the existence of at least one non-trivial positive solution for the critical Hénon equation provided that α → 0 + and for a suitable value of λ, where Ω ′ is a bounded domain in hyperbolic space B N . Finally, by working in the hole hyperbolic space H N , He [12] considered the following Hardy-Hénon type system 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 [3] where they extended He's result by considering a general nonlinearity They were able to prove the existence of at least one positive solution through a compact Sobolev embedding with the Mountain Pass Theorem. In this paper we investigate the existence of infinitely many solutions by considering a gradient system that generalizes problem (1). We cite [2,4,7,8,9,14,15] for related gradient systems problems. In order to obtain our result, we applied the Clark's theorem [5,6] and get inspiration on the nonlinearities condition employed by Morais Filho and Souto [13] in a p-laplacian system defined on a bounded domain in R N .
As regarding the difficulties, many technical difficulties arise when working on B N , which is a non compact manifold. This means that the the embedding H 1 (B N ) ֒→ L p (B N ) is not compact for 2 ≤ p ≤ 2N N −2 and the functional related to the system H cannot satisfy the (P S) c condition for all c > 0.
We also point out that since the weight function d(x) depends on the Riemannian distance r from a pole o, we have some difficulties in proving that leading to a great effort in proving that the Euler-Lagrange functional associated is well defined.
To overcome these difficulties we restrict ourselves to the radial functions. Our result is , the problem (H) has infinitely many solutions.

Preliminaries
Throughout this paper, C is a positive constant which may change from line to line. The Poincaré ball for the hyperbolic space is endowed with Riemannian metric g given by g i,j = (p(x)) 2 δ i,j where p(x) = 2 1 − |x| 2 . We denote the hyperbolic volume by dV B N = (p(x)) N dx. The geodesic distance from the origin to x ∈ B N is given by The hyperbolic gradient and the Laplace-Beltrami operator are where H 1 (B N ) denotes the Sobolev space on B N with the metric g. ∇ and div denote the Euclidean gradient and divergence in R N , respectively.
: u is radial}. We shall find weak-solutions of problem (H) in the space endowed with the norm One can observe that system (H) is formally derived as the Euler-Lagrange equation for the functional We endowed the norm for To solve this problem we need the following lemmas.
Proof. In [10, Lemma 2.2] it has been proved that the map Applying the Cauchy's inequality ab ≤ a 2 +b 2 2 , we get By the subaditivity, and the lemma holds.
Remark 2.1. From the previous lemma, there exists a positive constant C > 0, such that where m = β p and 2 < p < , that is, 2 < p < δ.
Proof. Let (u n , v n ) ∈ H be a bounded sequence. Then up to a subsequence, if necessary, we may assume that It is easy to sea that u n ⇀ u and v n ⇀ v in H 1 r (B N ). We will use the same calculus used by Haiyang He [10] (page 26). We want to show that lim Let u ∈ H 1 r (B N ), then by Haiyang He [10] we have Since {|x| ≤ 1 2 }, ln 1 + |x| 1 − |x| ≤ 2r 1 − r 2 and 2 < p <m, we have Since {|x| > 1 2 } and 2 < p <m, we have By the Dominated convergence theorem, we obtain In the same way we conclude that and the Lemma holds.

Proof of Theorem 1.1
The Clark's theorem is one of the most important results in critical point theory (see [5]). It was successfully applied to sublinear elliptic problems with symmetry and the existence of infinitely many solutions around the 0 was shown. In order to state the Clark's theorem, we need some terminologies. Let (X, · X ) be a Banach Space and I ∈ C 1 (X, R).
(i) For c ∈ R we say that I(u) satisfies (P S) c condition if any sequence (u j ) ∞ j=1 ⊂ X such that I(u j ) → c and I ′ (u j ) → 0 has a convergent subsequence. (iii) Let Ω be a open and bounded set, 0 ∈ Ω in R n . If A ∈ S is such that there exists a odd homeomorphism function from A to ∂Ω, then γ(A) = n.

Theorem 3.1 (Clark's Theorem).
Let I ∈ C(X, R) be an even function, bounded from below, with I(0) = 0 and there exists a compact, symmetric set K ∈ S such that γ(K) = k and sup K I < 0. Then I has least k distinct pairs of critical points.
The proof of Theorem 1.1 is made by using Theorem 3.1. The (H) system are the Euler-Lagrange equations related to the functional The functional I is not bounded from below, therefore, we can't apply the Clark's technique for this functional.
In order to overcome this difficulty we consider the auxiliary functional We will show that the set of critical points of J is related to a set of critical points of I and J satisfies the conditions of Theorem 3.1.
The proof of Theorem 1.1 is divided into several lemmas.
Using the p − 1-homogeneity condition of Q u (u, v) and Q u (u, v), observe that Hence (w, z) is a weak solution for problem (H) and so, a critical point for I.
we conclude that (u n , v n ) is bounded. So, there exists (u, v) ∈ H such that, passing to a subsequence if necessary, (u n , v n ) ⇀ (u, v), as n → ∞.
From the Embedding Lemma 2.2, and by (K 2 ) − (Q 2 ), we infer that Therefore, by the Lebesgue Dominated Convergence Theorem, then, (u n , v n ) → (u, v) . Therefore, The next lemma ends the proof of the Theorem 1.1.

Lemma 3.3.
Given k ∈ N, there exists a compact and symmetric set K ∈ H such that γ(K) = k and sup K J < 0.
Proof. Let X k ⊂ H be a subspace of dimension k. Consider the following norm in X k Since X k ⊂ H has finite dimension, there exists a > 0 such that Therefore, we obtain from (Q 3 ) that where C ∈ R is a positive constant. We then conclude that We get that sup K J < 0, where K ⊂ H is a compact and symmetric set such that γ(K) = k.
Finally, from Lemmas 3.2 and 3.3, Theorem 3.1 implies the existence of at least k distinct pairs of critical points for the functional J. Since k is arbitrary, we obtain infinitely many critical points for J.
In view of Lemma 3.1, we conclude that the functional J possesses, together with I, infinitely many critical points.
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 [1,3], we can conclude that (u, v) is a critical point in

Further result
We can apply the same method that proved Theorem 1.1 to establish the existence of infinitely many solutions for the following semilinear elliptic equation We obtain the following result The energy functional corresponding to (H * ) is defined on H 1 r (B N ) Problem (H * ) is close related to one studied by Carrião, Faria and Miyagaki [3]. In [3], they proved that the map and then there exists a positive constant C > 0, such that by taking m = β q with 2 < q < 2N N −2−2 β q +α .
Using (K 1 )-(K 2 ) together with inequality (6), we get that the functional I is well defined. This functional is not bounded from below, hence, we can't apply Clark's technique [5].
In order to overcome this difficulty we consider the auxiliary functional where p ∈ (2, 2 β α ), u ∈ E and We have the corresponding results of Lemmas 3.1, 3.2 and 3.3 for problem (H * ). The set of critical points of ψ is related to a set of critical points of I and ψ satisfies the conditions of Theorem 3.1.  Given k ∈ N, there exists a compact and symmetric set K ∈ E such that γ(K) = k and sup K ψ < 0.
From Lemmas 4.2 and 4.3 and Theorem 3.1, we conclude that the functional I possesses infinitely many critical points.