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

  • Patrícia Leal da CunhaEmail author
  • Flávio Almeida Lemos
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Abstract

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

Keywords

Hyperbolic space Hénon equation Variational methods 

MSC

58J05 35J60 58E30 35J20 35J75 

1 Introduction and the main result

This article concerns the existence of infinitely many solutions for the following semilinear elliptic system of Hénon type in hyperbolic space:
where \(\mathbb{B}^{N} \) is the Poincaré ball model for the hyperbolic space, \(H_{r}^{1}(\mathbb{B}^{N})\) denotes the Sobolev space of a radial \(H^{1}(\mathbb{B}^{N})\) function, \(r=d(x)=d_{\mathbb{B} ^{N}}(0,x)\), \(\Delta _{\mathbb{B}^{N}}\) is the Laplace–Beltrami type operator on \(\mathbb{B}^{N}\).
We assume the following hypotheses on K and Q:
\((K_{1})\)

\(K\geq 0\) is a continuous function with \(K(0)=0\) and \(K(t)\neq 0\) for \(t\neq 0\).

\((K_{2})\)

\(K=O(r^{\beta })\) as \(r \to 0\) and \(K=O(r^{\beta })\) as \(r \to \infty \) for some \(\beta >0\).

\((Q_{1})\)
\(Q \in C^{1}(\mathbb{R}\times \mathbb{R},\mathbb{R})\) is such that \(Q(-s,t)= Q(s,-t)=Q(s,t)\), \(Q(\lambda s, \lambda t)= \lambda ^{p} Q(s,t)\) (Q is p-homogeneous), \(\forall \lambda \in \mathbb{R}\) and \(p\in (2,\delta )\), where
$$\delta = \textstyle\begin{cases} \frac{2N + 2\beta }{N-2 } & \text{if } N-2>0, \\ \infty & \text{if } \text{otherwise}. \end{cases} $$
\((Q_{2})\)

There exist \(C,C_{1} , C_{2} >0\) such that \(Q (s,t)\leq C(s^{p} + t^{p})\), \(Q_{s} (s,t)\leq C_{1} s^{p-1}\), and \(Q_{t} (s,t)\leq C_{2} t^{p-1}\), \(\forall s,t \geq 0\).

\((Q_{3})\)

There exists \(C_{3} >0\) such that \(C_{3}(|s|^{p}+ |t|^{p}) \leq Q(s,t)\) with \(p\in (2,\delta )\).

In the past few years the prototype problem
$$ -\Delta _{\mathbb{B}^{N}}u = d(x)^{\alpha } \vert u \vert ^{p-2}u, \quad u\in H _{r}^{1}\bigl( \mathbb{B}^{N}\bigr) $$
has attracted attention. Unlike the corresponding problem in the Euclidean space \(\mathbb{R}^{N}\), He in [1] proved the existence of a positive solution to the above problem over the range \(p\in (2,\frac{2N+2\alpha }{N-2})\) 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 [2], He and Qiu proved the existence of at least one non-trivial positive solution for the critical Hénon equation
$$ -\Delta _{\mathbb{B}^{N}}u = d(x)^{\alpha } \vert u \vert ^{2^{*}-2}u+\lambda u, \quad u\geq 0, u\in H_{0}^{1} \bigl(\varOmega '\bigr), $$
provided that \(\alpha \rightarrow 0^{+}\) and for a suitable value of λ, where \(\varOmega '\) is a bounded domain in hyperbolic space \(\mathbb{B}^{N}\). Finally, by working in the whole hyperbolic space \(\mathbb{H}^{N}\), He [3] considered the following Hardy–Hénon type system:
$$ \textstyle\begin{cases} -\Delta _{\textbf{H}^{N}}u= d_{b}(x)^{\alpha } \vert v \vert ^{p-1}v, \\ -\Delta _{\textbf{H}^{N}}v= d_{b}(x)^{\beta } \vert u \vert ^{q-1}u \end{cases} $$
for \(\alpha , \beta \in \mathbb{R}\), \(N>4\) 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
$$ \textstyle\begin{cases} -\Delta _{\mathbb{B}^{N}}^{\alpha }u= K(d(x)) f(u) \\ \quad u\in H_{r}^{1}(\mathbb{B}^{N}). \end{cases} $$
(1)
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 [5, 6, 7, 8, 9, 10, 11] for related gradient system problems. In order to obtain our result, we applied Clark’s theorem [12, 13] and got inspiration on the nonlinearities condition employed by Morais Filho and Souto [14] in a p-Laplacian system defined on a bounded domain in \(\mathbb{R}^{N}\).

Regarding the difficulties, many technical difficulties arise when working on \(\mathbb{B}^{N}\), which is a non-compact manifold. This means that the embedding \(H^{1}(\mathbb{B}^{N})\hookrightarrow L^{p}( \mathbb{B}^{N})\) is not compact for \(2\leq p\leq \frac{2N}{N-2}\) and the functional related to the system \(\mathcal {H}\) cannot satisfy the \((PS)_{c}\) 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 0, we have some difficulties in proving that
$$ \int _{\mathbb{B}^{N}}d(x)^{\beta } \bigl( \bigl\vert u(x) \bigr\vert ^{p} + \bigl\vert v(x) \bigr\vert ^{p} \bigr) \,dV_{\mathbb{B}^{N}} < \infty , \quad \forall (u,v)\in H^{1}\bigl( \mathbb{B}^{N}\bigr)\times H^{1}\bigl( \mathbb{B}^{N}\bigr) $$
leading to a great effort in proving that the associated Euler–Lagrange functional is well defined.

To overcome these difficulties, we restrict ourselves to the radial functions.

Our result is the following.

Theorem 1.1

Under hypotheses\((K_{1})\)\((K_{2})\)and\((Q_{1})\)\((Q_{3})\), problem (\(\mathcal {H}\)) has infinitely many solutions.

2 Preliminaries

Throughout this paper, C is a positive constant which may change from line to line.

The Poincaré ball for the hyperbolic space is
$$\mathbb{B}^{N}=\bigl\{ x \in \mathbb{R}^{N}| \vert x \vert < 1\bigr\} $$
endowed with Riemannian metric g given by \(g_{i,j}=(p(x))^{2} \delta _{i,j} \), where \(p(x)= \frac{2}{1-|x|^{2}}\). We denote the hyperbolic volume by \(dV_{\mathbb{B}^{N}}=(p(x))^{N} \,dx\). The hyperbolic distance from the origin to \(x \in \mathbb{B}^{N}\) is given by
$$d(x):=d_{\mathbb{B}^{N}}(0,x)= \int _{0}^{ \vert x \vert }\frac{2}{1-s^{2}}\,ds= \log \biggl(\frac{1+ \vert x \vert }{1- \vert x \vert } \biggr). $$
The hyperbolic gradient and the Laplace–Beltrami operator are
$$-\Delta _{\mathbb{B}^{N}}u=-\bigl(p(x)\bigr)^{-N}\operatorname{div} \bigl(p(x)^{N-2} \nabla u\bigr), \quad \nabla _{\mathbb{B}^{N}}u= \frac{\nabla u}{p(x)}, $$
where \(H^{1}(\mathbb{B}^{N})\) denotes the Sobolev space on \(\mathbb{B}^{N}\) with the metric g. ∇ and div denote the Euclidean gradient and divergence in \(\mathbb{R}^{N}\), respectively.

Let \(H^{1}_{r}(\mathbb{B}^{N})=\{ u \in H^{1}(\mathbb{B}^{N}):u \text{ is radial} \}\).

We shall find weak solutions of problem (\(\mathcal {H}\)) in the space
$$H= H_{r}^{1}\bigl(\mathbb{B}^{N}\bigr) \times H_{r}^{1}\bigl(\mathbb{B}^{N}\bigr) $$
endowed with the norm
$$\bigl\lVert (u,v) \bigr\rVert ^{2}= \int _{\mathbb{B}^{N}} \bigl( \vert {\nabla _{\mathbb{B}^{N}}u} \vert _{\mathbb{B}^{N}}^{2}+ \vert {\nabla _{\mathbb{B}^{N}}v} \vert _{ \mathbb{B}^{N}}^{2} \bigr)\,dV_{\mathbb{B}^{N}}. $$
One can observe that system (\(\mathcal {H}\)) is formally derived as the Euler–Lagrange equation for the functional
$$ I(u,v)= \frac{1}{2} \int _{\mathbb{B}^{N}} \bigl( \vert {\nabla _{\mathbb{B}^{N}}u} \vert _{ \mathbb{B}^{N}}^{2}+ |{\nabla _{\mathbb{B}^{N}}v|}_{\mathbb{B}^{N}} ^{2}\bigr)\,dV_{\mathbb{B}^{N}}- \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr)Q(u,v)\,dV _{\mathbb{B}^{N}}. $$
We endowed the norm for \(L^{p}(\mathbb{B}^{N})\times L^{p}(\mathbb{B} ^{N})\) as follows:
$$\bigl\lVert (u,v) \bigr\rVert _{p}^{p}= \int _{\mathbb{B}^{N}}\bigl( \vert u \vert ^{p}+ \vert v \vert ^{p}\bigr)\,dV_{\mathbb{B}^{N}}. $$

To solve this problem, we need the following lemmas.

Lemma 2.1

The map\((u,v) \longmapsto (d(x)^{m} u,d(x)^{m} v) \)from\(H= H_{r} ^{1} (\mathbb{B}^{N})\times H_{r}^{1}(\mathbb{B}^{N})\)to\(L^{p}( \mathbb{B}^{N})\times L^{p}(\mathbb{B}^{N})\)is continuous for\(p \in (2,\tilde{m})\), where\(m>0\)and
$$ \tilde{m} = \textstyle\begin{cases} \frac{2N}{N-2-2m } & \textit{if } m < \frac{N-2 }{2}, \\ \infty & \textit{if otherwise}. \end{cases} $$

Proof

In [1, Lemma 2.2] it has been proved that the map \(u \longmapsto d(x)^{m} u \) from \(H_{r}^{1}(\mathbb{B}^{N})\) to \(L^{p}(\mathbb{B} ^{N})\) is continuous for \(p \in (2,\tilde{m})\). Therefore \(\lVert d(x)^{m}u \rVert _{p}\leq C \lVert u \rVert _{H^{1}_{r}}\) and \(\lVert d(x)^{m}v \rVert _{p}\leq C \lVert v \rVert _{H^{1}_{r}}\). Hence,
$$\begin{aligned} \bigl( \bigl\lVert d(x)^{m}u \bigr\rVert _{p}^{2}+ \bigl\lVert d(x)^{m}v \bigr\rVert _{p}^{2} \bigr)^{\frac{1}{2}}\leq C \bigl\lVert (u,v) \bigr\rVert . \end{aligned}$$
Now observe that
$$\begin{aligned} \bigl\lVert d(x)^{m}u \bigr\rVert _{p} + \bigl\lVert d(x)^{m}v \bigr\rVert _{p} =& \bigl[ \bigl( \bigl\lVert d(x)^{m}u \bigr\rVert _{p} + \bigl\lVert d(x)^{m}v \bigr\rVert _{p} \bigr)^{2} \bigr]^{\frac{1}{2}} \\ =& \bigl( \bigl\lVert d(x)^{m}u \bigr\rVert _{p}^{2} +2 \bigl\lVert d(x)^{m}u \bigr\rVert _{p} \bigl\lVert d(x)^{m}v \bigr\rVert _{p} + \bigl\lVert d(x)^{m}v \bigr\rVert _{p} \bigr)^{\frac{1}{2}}. \end{aligned}$$
Applying Cauchy’s inequality \(ab\leq \frac{a^{2} + b^{2}}{2}\), we get
$$\begin{aligned} \bigl\lVert d(x)^{m}u \bigr\rVert _{p} + \bigl\lVert d(x)^{m}v \bigr\rVert _{p} \leq \sqrt{2} \bigl( \bigl\lVert d(x)^{m}u \bigr\rVert _{p} ^{2}+ \bigl\lVert d(x)^{m}v \bigr\rVert _{p}^{2} \bigr)^{\frac{1}{2}}. \end{aligned}$$
By the subadditivity, we get
$$\begin{aligned} \bigl( \bigl\lVert d(x)^{m}u \bigr\rVert _{p}^{p}+ \bigl\lVert d(x)^{m}v \bigr\rVert _{p}^{p} \bigr)^{\frac{1}{p}}\leq \bigl\lVert d(x)^{m}u \bigr\rVert _{p} + \bigl\lVert d(x)^{m}v \bigr\rVert _{p}. \end{aligned}$$
Therefore,
$$\begin{aligned} \bigl\lVert \bigl(d(x)^{m}u,d(x)^{m}v\bigr) \bigr\rVert _{p}\leq C \bigl\lVert (u,v) \bigr\rVert , \end{aligned}$$
and the lemma holds. □

Remark 2.1

From the previous lemma, there exists a positive constant \(C>0\) such that
$$\begin{aligned}& \int _{\mathbb{B}^{N}}d(x)^{\beta } \bigl\vert u(x) \bigr\vert ^{p} \,dV_{\mathbb{B}^{N}} + \int _{\mathbb{B}^{N}}d(x)^{\beta } \bigl\vert v(x) \bigr\vert ^{p} \,dV_{\mathbb{B}^{N}} \\& \quad \leq C \biggl( \int _{\mathbb{B}^{N}} \vert {\nabla _{\mathbb{B}^{N}}u} \vert _{ \mathbb{B}^{N}}^{2}\,dV_{\mathbb{B}^{N}} + \int _{\mathbb{B}^{N}} \vert {\nabla _{ \mathbb{B}^{N}}v} \vert _{\mathbb{B}^{N}}^{2}\,dV_{\mathbb{B}^{N}} \biggr) ^{\frac{p}{2}}, \end{aligned}$$
where \(m=\frac{\beta }{p}\) and \(2< p< \frac{2N}{N-2-2(\frac{\beta }{p})}\), that is, \(2< p<\delta \).

Lemma 2.2

The map\((u,v) \longmapsto (d(x)^{m} u,d(x)^{m} v) \)from\(H= H_{r} ^{1}(\mathbb{B}^{N})\times H_{r}^{1}(\mathbb{B}^{N})\)to\(L^{p}( \mathbb{B}^{N})\times L^{p}(\mathbb{B}^{N})\)is compact for\(p \in (2,\tilde{m})\), where\(m>0\)and
$$\tilde{m} = \textstyle\begin{cases} \frac{2N}{N-2-2m } & \textit{if } m< \frac{N-2 }{2}, \\ \infty & \textit{if otherwise}. \end{cases} $$

Proof

Let \((u_{n}, v_{n}) \in H \) be a bounded sequence. Then, up to a subsequence, if necessary, we may assume that
$$(u_{n}, v_{n})\rightharpoonup (u,v). $$

It is easy to see that \(u_{n} \rightharpoonup u\) and \(v_{n} \rightharpoonup v\) in \(H_{r}^{1}(\mathbb{B}^{N})\).

We will use the same calculus used by Haiyang He [1] (page 26). We want to show that
$$\begin{aligned}& \lim_{n\to \infty } \int _{\mathbb{B}^{N}}d(x)^{mp} \bigl\vert u_{n}(x) \bigr\vert ^{p} \,dV _{\mathbb{B}^{N}} = \int _{\mathbb{B}^{N}}d(x)^{mp} \bigl\vert u(x) \bigr\vert ^{p} \,dV_{ \mathbb{B}^{N}}, \\& \lim_{n\to \infty } \int _{\mathbb{B}^{N}}d(x)^{mp} \bigl\vert v_{n}(x) \bigr\vert ^{p} \,dV _{\mathbb{B}^{N}} = \int _{\mathbb{B}^{N}}d(x)^{mp} \bigl\vert v(x) \bigr\vert ^{p} \,dV_{ \mathbb{B}^{N}}. \end{aligned}$$
Let \(u\in H_{r}^{1}(\mathbb{B}^{N})\), then by Haiyang He [1] we have
$$\begin{aligned}& \bigl\vert u_{n}(x) \bigr\vert \leq \frac{1}{\sqrt{\omega _{n-1}(N-2)}} \biggl(\frac{1- \vert x \vert ^{2}}{2} \biggr)^{\frac{N-2}{2}}\frac{1}{ \vert x \vert ^{\frac{N-2}{2}}} \lVert u_{n} \rVert _{H_{r}^{1}(\mathbb{B}^{N})}, \\& \bigl\vert u_{n}(x) \bigr\vert \leq \frac{1}{\sqrt{\omega _{n-1}}} \biggl(\frac{1- \vert x \vert ^{2}}{2} \biggr) ^{\frac{N-1}{2}}\frac{1}{ \vert x \vert ^{\frac{N}{2}}} \lVert u_{n} \rVert _{H_{r}^{1}(\mathbb{B}^{N})}. \end{aligned}$$
Since \(\{|x|\leq \frac{1}{2}\}\), \(\ln \frac{1+ |x|}{1-|x|}\leq \frac{2r}{1-r ^{2}}\), and \(2< p< \tilde{m}\), we have
$$\begin{aligned} d(x)^{mp} \vert u \vert ^{p} \leq& C \biggl(\ln \frac{1+ \vert x \vert }{1- \vert x \vert } \biggr)^{mp} \biggl(\frac{1- \vert x \vert ^{2}}{2} \biggr)^{p\frac{N-2}{2}} \biggl(\frac{1}{ \vert x \vert ^{ \frac{N-2}{2}}} \biggr)^{p} \\ \leq& C \biggl(\frac{2 \vert x \vert }{1- \vert x \vert ^{2}} \biggr)^{mp} \biggl( \frac{1- \vert x \vert ^{2}}{2} \biggr) ^{p\frac{N-2}{2}} \biggl(\frac{1}{ \vert x \vert ^{\frac{N-2}{2}}} \biggr)^{p} \equiv h_{1}. \end{aligned}$$
Set
$$g_{1}(x) = \textstyle\begin{cases} h_{1}(x) & \text{if } 0\leq \vert x \vert < \frac{1}{2}, \\ 0 & \text{if }\frac{1}{2}\leq \vert x \vert < 1, \end{cases} $$
then
$$\begin{aligned} \int _{\mathbb{B}^{N}}g_{1} \,dV_{\mathbb{B}^{N}} =& \int _{0}^{ \frac{1}{2}} \biggl(\frac{2r}{1-r^{2}} \biggr)^{mp} \biggl(\frac{1- r ^{2}}{2} \biggr)^{p\frac{N-2}{2}} \biggl(\frac{1}{r^{\frac{N-2}{2}}} \biggr) ^{p}r^{N-1} \biggl( \frac{2}{1- r^{2}} \biggr)^{N}\,dr \\ \leq& C \int _{0}^{\frac{1}{2}} \biggl(\frac{2r}{1-r^{2}} \biggr)^{mp} \biggl(\frac{1- r^{2}}{2} \biggr)^{(p\frac{N-2}{2} - N)}r^{N-1 - p \frac{N-2}{2}} \,dr \\ \leq &C \int _{0}^{\frac{1}{2}}r^{mp +N -1 -p\frac{N-2}{2}}\,dr < \infty . \end{aligned}$$
Since \(\{|x|> \frac{1}{2}\}\) and \(2< p< \tilde{m}\), we have
$$\begin{aligned} d(x)^{mp} \vert u \vert ^{p} \leq& C \biggl(\ln \frac{1+ \vert x \vert }{1- \vert x \vert } \biggr)^{mp} \biggl(\frac{1- \vert x \vert ^{2}}{2} \biggr)^{p\frac{N-1}{2}} \biggl(\frac{1}{ \vert x \vert ^{ \frac{N}{2}}} \biggr)^{p} \\ \leq& C \biggl(\ln \frac{1+ \vert x \vert }{1- \vert x \vert } \biggr)^{mp} \biggl( \frac{1- \vert x \vert ^{2}}{2} \biggr)^{p\frac{N-1}{2}} \biggl(\frac{1}{ \vert x \vert ^{ \frac{N}{2}}} \biggr)^{p} \equiv h_{2}. \end{aligned}$$
Set
$$g_{2}(x) = \textstyle\begin{cases} 0 & \text{if } 0\leq \vert x \vert < \frac{1}{2} \\ h_{2}(x) & \text{if }\frac{1}{2}\leq \vert x \vert < 1, \end{cases} $$
then
$$\begin{aligned} \int _{\mathbb{B}^{N}}g_{2} \,dV_{\mathbb{B}^{N}} =& \int _{\frac{1}{2}} ^{1} \biggl(\ln \frac{1+ r}{1-r} \biggr)^{mp} \biggl(\frac{1- r^{2}}{2} \biggr) ^{p\frac{N-1}{2}} \biggl(\frac{1}{r^{\frac{N}{2}}} \biggr)^{p}r^{N-1} \biggl( \frac{2}{1- r^{2}} \biggr)^{N}\,dr \\ \leq& C \int _{\frac{1}{2}}^{1} \biggl(\ln \frac{1+ r}{1-r} \biggr) ^{mp} \biggl(\frac{1- r^{2}}{2} \biggr)^{(p\frac{N-1}{2} - N)}r^{N-1 - \frac{N}{2}p} \,dr \\ \leq& \int _{\ln 3}^{\infty }s^{mp} \biggl( \frac{2e^{s}}{(e^{s} +1)^{2}} \biggr) ^{\frac{N-1}{2}p-N +1}\,ds< \infty . \end{aligned}$$
Hence, we have
$$\bigl\vert d(x)^{mq}u_{n}(x)^{q} \bigr\vert \leq g_{1}(x) + g_{2}(x). $$
By the dominated convergence theorem, we obtain
$$\lim_{n\to \infty } \int _{\mathbb{B}^{N}}d(x)^{mq}u_{n}^{p}(x) \,dV_{ \mathbb{B}^{N}} = \int _{\mathbb{B}^{N}}d(x)^{mp}u^{p}(x) \,dV_{ \mathbb{B}^{N}}. $$
In the same way we conclude that
$$\lim_{n\to \infty } \int _{\mathbb{B}^{N}}d(x)^{mq}v_{n}^{p}(x) \,dV_{ \mathbb{B}^{N}} = \int _{\mathbb{B}^{N}}d(x)^{mp}v^{p}(x) \,dV_{ \mathbb{B}^{N}} , $$
and the lemma holds. □

3 Proof of Theorem 1.1

Clark’s theorem is one of the most important results in critical point theory (see [12]). It was successfully applied to sublinear elliptic problems with symmetry and the existence of infinitely many solutions around was shown.

In order to state Clark’s theorem, we need some terminologies.

Let \((X, \lVert \cdot \rVert _{X})\) be a Banach space and \(\mathcal{I} \in C^{1}(X,\mathbb{R})\).
  1. (i)

    For \(c \in \mathbb{R}\), we say that \(\mathcal{I}(u)\) satisfies the \((PS)_{c}\) condition if any sequence \((u_{j})_{j=1}^{ \infty } \subset X\) such that \(\mathcal{I}(u_{j})\to c\) and \(\lVert \mathcal{I}'(u_{j}) \rVert \to 0\) has a convergent subsequence.

     
  2. (ii)

    Let S be a symmetric and closed set family in \(X\diagdown \{0\}\). For \(A \in S\), the genus \(\gamma (A)=\min \{n \in \mathbb{N}:\phi \in C(A, \mathbb{R}^{n} \{0\}) \text{ is odd}\}\). If there is no such natural, we set \(\gamma (A)= \infty \).

     
  3. (iii)

    Let Ω be an open and bounded set, \(0\in \varOmega \) in \(\mathbb{R}^{n}\). If \(A\in S\) is such that there exists an odd homeomorphism function from A to ∂Ω, then \(\gamma (A)=n\).

     

Theorem 3.1

(Clark’s theorem)

Let\(\mathcal{I} \in C(X, \mathbb{R})\)be an even function bounded from below with\(\mathcal{I}(0)=0\), and there exists a compact, symmetric set\(K \in X\)such that\(\gamma (K)=k\)and\(\sup_{K}\mathcal{I}<0\). ThenIhas at leastkdistinct pairs of critical points.

The proof of Theorem 1.1 is made by using Theorem 3.1.

The (\(\mathcal {H}\)) system is the Euler–Lagrange equations related to the functional
$$ I(u,v)= \frac{1}{2} \int _{\mathbb{B}^{N}}\bigl( \vert {\nabla _{\mathbb{B}^{N}}u} \vert _{ \mathbb{B}^{N}}^{2}+ \vert {\nabla _{\mathbb{B}^{N}}v} \vert _{\mathbb{B}^{N}} ^{2}\bigr)\,dV_{\mathbb{B}^{N}}- \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr)Q(u,v)\,dV_{ \mathbb{B}^{N}}, $$
(2)
which is \(C^{1}\) on H.

The functional I is not bounded from below, therefore, we cannot apply Clark’s technique for this functional.

In order to overcome this difficulty, we consider the auxiliary functional
$$ J(u,v)= \biggl( \int _{\mathbb{B}^{N}}\bigl( \vert {\nabla _{\mathbb{B}^{N}}u} \vert _{ \mathbb{B}^{N}}^{2}+ \vert {\nabla _{\mathbb{B}^{N}}v} \vert _{\mathbb{B}^{N}} ^{2}\bigr)\,dV_{\mathbb{B}^{N}} \biggr)^{p-1} - \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr)Q(u,v) \,dV_{\mathbb{B}^{N}}, $$
(3)
where \(p \in (2, \delta )\), while for \(J'\) we have \(\forall (\phi , \psi )\in H\)
$$\begin{aligned} J'(u,v) (\phi , \psi ) =& (2p-2) \bigl\lVert (u,v) \bigr\rVert ^{2p-4} \int _{\mathbb{B}^{N}} \bigl( \langle \nabla _{\mathbb{B}^{N}}u, \nabla _{\mathbb{B}^{N}}\phi \rangle _{\mathbb{B}^{N}}+ \langle \nabla _{\mathbb{B}^{N}}v,\nabla _{\mathbb{B}^{N}}\psi \rangle _{ \mathbb{B}^{N}}\bigr) \,dV_{\mathbb{B}^{N}} \\ &{}- \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \bigl(\phi Q_{u}(u,v) + \psi Q_{v}(u,v)\bigr) \,dV_{\mathbb{B}^{N}}. \end{aligned}$$
(4)

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.

Lemma 3.1

If\((u,v)\in H\), \((u,v) \neq (0,0)\)is a critical point forJ, then
$$(w,z)= \biggl(\frac{u}{[(2p-2) \lVert (u,v) \rVert ^{2p-4}]^{ \frac{1}{p-2}}}, \frac{v}{[(2p-2) \lVert (u,v) \rVert ^{2p-4}]^{\frac{1}{p-2}}} \biggr) $$
is a critical point forI.

Proof

Note that \((u,v)\neq (0,0)\) is a critical point for J if, and only if, \((u,v)\) is a weak solution to the problem

Define \(\lambda ( \lVert (u,v) \rVert )= [(2p-2) \lVert (u,v) \rVert ^{2p-4} ]^{\frac{-1}{p-2}}\), then \((w,z)=\lambda ( \lVert (u,v) \rVert ) (u,v) \).

Using the \(p-1\)-homogeneity condition of \(Q_{u}(u,v)\) and \(Q_{v}(u,v)\), observe that
$$\begin{aligned}& - \Delta _{\mathbb{B}^{N}}w-K\bigl(d(x)\bigr)Q_{u}(w,z) \\& \quad = -\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr)\Delta _{\mathbb{B} ^{N}}u - \bigl(\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr) \bigr)^{p-1}K\bigl(d(x)\bigr)Q _{u}(u,v) \\& \quad = -\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr) K\bigl(d(x)\bigr) Q_{u}(u,v) \biggl( \bigl(\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr)\bigr)^{p-2} - \frac{1}{(2p-2) \lVert (u,v) \rVert ^{2p-4}} \biggr) \end{aligned}$$
and
$$\begin{aligned}& - \Delta _{\mathbb{B}^{N}}z-K\bigl(d(x)\bigr)Q_{v}(w,z) \\& \quad = -\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr)\Delta _{\mathbb{B} ^{N}}v - \bigl(\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr) \bigr)^{p-1}K\bigl(d(x)\bigr)Q _{v}(u,v) \\& \quad = -\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr) K\bigl(d(x)\bigr) Q_{v}(u,v) \biggl( \bigl(\lambda \bigl( \bigl\lVert (u,v) \bigr\rVert \bigr)\bigr)^{p-2} - \frac{1}{(2p-2) \lVert (u,v) \rVert ^{2p-4}} \biggr). \end{aligned}$$
Hence \((w,z)\) is a weak solution for problem (\(\mathcal {H}\)) and so, a critical point for I. □

Lemma 3.2

\(J(u,v)\)is bounded from below and satisfies the\((PS)_{c}\)condition.

Proof

From \((K_{1})\)\((K_{2})\), \((Q_{2})\)\((Q_{3})\), and Remark 2.1
$$\begin{aligned} J(u,v) =& \biggl( \int _{\mathbb{B}^{N}} \vert {\nabla _{\mathbb{B}^{N}}u} \vert _{ \mathbb{B}^{N}}^{2}+ \vert {\nabla _{\mathbb{B}^{N}}v} \vert _{\mathbb{B}^{N}} ^{2}\,dV_{\mathbb{B}^{N}} \biggr)^{p-1} - \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr)Q(u,v) \,dV_{\mathbb{B}^{N}} \\ \geq & \bigl\lVert (u,v) \bigr\rVert ^{2p-2} - C \int _{\mathbb{B} ^{N}}d(x)^{\beta }\bigl( \vert u \vert ^{p} + \vert v \vert ^{p}\bigr)\,dV_{\mathbb{B}^{N}} \\ \geq & \bigl\lVert (u,v) \bigr\rVert ^{2p-2} - \bigl\lVert (u,v) \bigr\rVert ^{p}, \end{aligned}$$
so that \(J(u,v)\) is bounded from below.
Let \((u_{n},v_{n}) \in H\) be such that \(|J(u_{n},v_{n})|\leq C\) with \(C\in \mathbb{R}^{+}\), \(J'(u_{n},v_{n})\to 0\). Since
$$C\geq J(u_{n} ,v_{n})\geq \bigl\lVert (u_{n},v_{n}) \bigr\rVert ^{2p-2} - \bigl\lVert (u_{n},v_{n}) \bigr\rVert ^{p}, $$
we conclude that \(\lVert (u_{n},v_{n}) \rVert \) is bounded. So, there exists \((u,v) \in H\) such that, passing to a subsequence if necessary,
$$ (u_{n}, v_{n}) \rightharpoonup (u,v), \quad \text{as } n \rightarrow \infty . $$
From the embedding Lemma 2.2, we have
$$\int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\beta }\bigl( \vert u_{n} \vert ^{p} + \vert v_{n} \vert ^{p}\bigr) \,dV _{\mathbb{B}^{N}}\longrightarrow \int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\beta }\bigl( \vert u \vert ^{p} + \vert v \vert ^{p} \bigr) \,dV_{\mathbb{B}^{N}}, $$
and by \((K_{2})-(Q_{2})\), we infer that
$$ |K(d(x) \bigl(u_{n}Q_{u}(u_{n},v_{n}) +v_{n}Q_{v}(u_{n},v_{n}) \bigr)|\leq C\bigl(d(x)\bigr)^{ \beta }\bigl( \vert u_{n} \vert ^{p}+ \vert v_{n} \vert ^{p} \bigr). $$
Therefore, by the Lebesgue dominated convergence theorem,
$$\begin{aligned}& \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \bigl( Q_{u}(u_{n} , v_{n})u_{n} +Q_{v}(u_{n},v _{n})v_{n} \bigr)\,dV_{\mathbb{B}^{N}}\\& \quad \rightarrow \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \bigl(Q _{u}(u,v)u +Q_{v}(u,v )v\bigr)\,dV_{\mathbb{B}^{N}}. \end{aligned}$$
Since \(J'(u,v)(u,v)=0\) and \(J'(u_{n},v_{n})(u_{n}, v_{n})=o_{n}(1)\) as \(n\to \infty \), we have
$$\begin{aligned}& (2p-2) \bigl( \bigl\lVert (u_{n},v_{n}) \bigr\rVert ^{2p-2}- \bigl\lVert (u,v) \bigr\rVert ^{2p-2} \bigr) \\& \quad =J'(u_{n},v_{n}) (u_{n}, v_{n})-J'(u,v) (u,v) \\& \qquad {}+ \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \bigl(Q_{u}(u_{n} , v_{n})u_{n} +Q_{v}(u _{n},v_{n})v_{n} \bigr) \,dV_{\mathbb{B}^{N}} \\& \qquad {}- \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \bigl(Q_{u}(u,v)u +Q_{v}(u,v )v \bigr)\,dV _{\mathbb{B}^{N}}= o_{n}(1), \end{aligned}$$
then \(\lVert (u_{n},v_{n}) \rVert \rightarrow \lVert (u,v) \rVert \). Therefore,
$$ (u_{n},v_{n})\rightarrow (u,v), \quad \text{as }n \rightarrow \infty ,\text{ in } H. $$
 □

The next lemma ends the proof of Theorem 1.1.

Lemma 3.3

Given\(k \in \mathbb{N}\), there exists a compact and symmetric set\(K \in H\)such that\(\gamma (K)=k\)and\(\sup_{K}J <0\).

Proof

Let \(X_{k} \subset H\) be a subspace of dimension k. Consider the following norm in \(X_{k}\):
$$\bigl\lVert (u,v) \bigr\rVert _{X_{k}}= \biggl( \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \bigl( \vert u \vert ^{p} + \vert v \vert ^{p}\bigr) \,dV_{\mathbb{B}^{N}} \biggr) ^{\frac{1}{p}}. $$
Since \(X_{k} \subset H\) has finite dimension, there exists \(a>0\) such that
$$a \bigl\lVert (u,v) \bigr\rVert _{X_{k}}\leq \bigl\lVert (u,v) \bigr\rVert \leq \frac{1}{a} \bigl\lVert (u,v) \bigr\rVert _{X_{k}}, \quad \forall (u,v)\in X_{k}. $$
Therefore, we obtain from \((Q_{3})\) that
$$J(u,v)\leq \bigl\lVert (u,v) \bigr\rVert ^{2p-2}-C \int _{\mathbb{B} ^{N}}K\bigl(d(x)\bigr) \bigl( \vert u \vert ^{p}+ \vert v \vert ^{p}\bigr)\,dV_{\mathbb{B}^{N}} = \bigl\lVert (u,v) \bigr\rVert ^{2p-2} - C \bigl\lVert (u,v) \bigr\rVert _{X_{k}}^{p}, $$
where \(C\in \mathbb{R}\) is a positive constant. We then conclude that
$$J(u,v)\leq \bigl\lVert (u,v) \bigr\rVert _{X_{k}}^{p} \biggl( \frac{ \lVert (u,v) \rVert _{X_{k}}^{p-2}}{a^{2p-2}}- C \biggr). $$
Let \(A=a^{\frac{2p-2}{p-2}}\) and consider the set \(K=\{ (u,v)\in X _{k}: \lVert (u,v) \rVert _{X_{k}}= \frac{A}{2}C^{ \frac{1}{p-2}} \}\), then
$$J(u,v)\leq C \frac{A^{p}}{2^{p}} \biggl( \frac{1}{2^{q-2}}-1 \biggr)< 0, \quad \forall (u,v) \in K. $$

We get that \(\sup_{K}J <0\), where \(K\subset H\) is a compact and symmetric set such that \(\gamma (K) =k\). □

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 in H.

In view of Lemma 3.1, we conclude that the functional J possesses, together with I, infinitely many critical points in H.

Finally, we point out that since H is a closed subspace of the Hilbert space \(H^{1}(\mathbb{B}^{N})\times H^{1}(\mathbb{B}^{N})\), following some ideas in [4, 15], we can conclude that \((u,v)\) is a critical point in \(H^{1}(\mathbb{B}^{N})\times H^{1}(\mathbb{B}^{N})\).

4 Further result

We can apply the same method used in the proof of Theorem 1.1 to establish the existence of infinitely many solutions for the following semilinear elliptic equation:
where K satisfies \((K_{1})-(K_{2})\), \(-\Delta _{\mathbb{B}^{N}}^{ \alpha }\) is the Laplace–Beltrami type operator
$$-\Delta _{\mathbb{B}^{N}}^{\alpha }u=-\bigl(p(x)\bigr)^{-N} \operatorname{div}\bigl(p(x)^{N-2}\bigl(d(x)\bigr)^{ \alpha } \nabla u\bigr) $$
and
$$\begin{aligned} \mathbb{E}= \biggl\{ u \in H^{1}_{r}\bigl( \mathbb{B}^{N}\bigr): \lVert u \rVert _{\mathbb{E}} = \biggl( \int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\alpha } \vert {\nabla _{ \mathbb{B}^{N}}u} \vert ^{2}_{\mathbb{B}^{N}} \,dV_{\mathbb{B}^{N}} \biggr) ^{\frac{1}{2}} < \infty \biggr\} . \end{aligned}$$

We obtain the following result.

Theorem 4.1

Under hypotheses\((K_{1})\)\((K_{2})\), (\(\mathcal {H*}\)) equation has infinitely many solutions.

The energy functional corresponding to (\(\mathcal {H*}\)) is
$$ I(u)= \frac{1}{2} \int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\alpha } \vert {\nabla _{\mathbb{B} ^{N}}u} \vert _{\mathbb{B}^{N}}^{2} \,dV_{\mathbb{B}^{N}}- \frac{1}{q} \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \vert u \vert ^{p}\,dV_{\mathbb{B}^{N}} $$
(5)
defined on E.
Problem (\(\mathcal {H*}\)) is closely related to the one studied by Carrião, Faria, and Miyagaki [4]. In [4], they proved that the map \(u \longmapsto d(x)^{m} u \) from \(\mathbb{E}\) to \(L^{q}(\mathbb{B}^{N})\) is compact for \(q \in (2,\tilde{m})\), where
$$\tilde{m} = \textstyle\begin{cases} \frac{2N}{N-2-2m + \alpha } & \text{if } m< \frac{N-2 + \alpha }{2}, \\ \infty & \text{if } \text{otherwise} \end{cases} $$
and then there exists a positive constant \(C>0\) such that
$$\begin{aligned} \int _{\mathbb{B}^{N}}d(x)^{\beta } \bigl\vert u(x) \bigr\vert ^{q} \,dV_{\mathbb{B}^{N}} \leq C \biggl( \int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\alpha } \vert {\nabla _{\mathbb{B} ^{N}}u} \vert _{\mathbb{B}^{N}}\,dV_{\mathbb{B}^{N}} \biggr)^{\frac{q}{2}}, \end{aligned}$$
(6)
by taking \(m=\frac{\beta }{q}\) with \(2< q<\frac{2N}{N-2-2 \frac{\beta }{q}+ \alpha }\).

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 cannot apply Clark’s technique [12].

In order to overcome this difficulty, we consider the auxiliary functional
$$ \psi (u)= \biggl( \int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\alpha } \vert {\nabla _{\mathbb{B} ^{N}}u} \vert _{\mathbb{B}^{N}}^{2} \,dV_{\mathbb{B}^{N}} \biggr)^{p-1} - \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \vert u \vert ^{p}\,dV_{\mathbb{B}^{N}}, $$
(7)
where \(p \in (2, 2_{\alpha }^{\beta })\), \(u\in \mathbb{E}\), and
$$\begin{aligned} \psi '(u)v =& (2p-2) \lVert u \rVert _{\mathbb{E}}^{2p-4} \int _{\mathbb{B}^{N}}\bigl(d(x)\bigr)^{\alpha } \langle \nabla _{\mathbb{B} ^{N}}u, \nabla _{\mathbb{B}^{N}}v \rangle _{\mathbb{B}^{N}}\,dV _{\mathbb{B}^{N}} \\ &{}- \int _{\mathbb{B}^{N}}K\bigl(d(x)\bigr) \vert u \vert ^{p-2}uv \,dV_{\mathbb{B}^{N}}. \end{aligned}$$
(8)

We have the corresponding results of Lemmas 3.1, 3.2, and 3.3 for problem (\(\mathcal {H*}\)). 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.

Lemma 4.1

If\(u\in \mathbb{E}\), \(u \neq 0\)is a critical point forψ, then\(v=\frac{u}{[(2p-2) \lVert u \rVert _{\mathbb{E}}^{2p-4}]^{ \frac{1}{p-2}}}\)is a critical point forI.

Lemma 4.2

\(\psi (u)\)is bounded from below and satisfies the Palais–Smale condition (PS).

Lemma 4.3

Given\(k \in \mathbb{N}\), there exists a compact and symmetric set\(K \in \mathbb{E}\)such that\(\gamma (K)=k\)and\(\sup_{K}\psi <0\).

From Lemmas 4.2 and 4.3 and Theorem 3.1, we conclude that the functional I possesses infinitely many critical points.

Notes

Acknowledgements

This paper was carried out while the first author was visiting the Mathematics Department of UFOP, and she would like to thank the members of DEMAT-UFOP for their hospitality. The authors are grateful to professor Giovany Figueiredo for helpful comments.

Availability of data and materials

Not applicable.

Authors’ contributions

All authors worked together to produce the results, read and approved the final manuscript.

Funding

Not applicable.

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    He, H.: The existence of solutions for Hénon equation in hyperbolic space. Proc. Jpn. Acad., Ser. A, Math. Sci. 89, 24–28 (2013) CrossRefGoogle Scholar
  2. 2.
    He, H., Qiu, J.: Existence of solutions for critical Hénon equations in hyperbolic spaces. Electron. J. Differ. Equ. 13, 5 (2013) zbMATHGoogle Scholar
  3. 3.
    He, H.: Infinitely many solutions for Hardy–Hénon type elliptic system in hyperbolic space. Ann. Acad. Sci. Fenn., Math. 40, 969–983 (2015) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carrião, P.C., Faria, L.F.O., Miyagaki, O.H.: Semilinear elliptic equations of the Hénon-type in hyperbolic space. Commun. Contemp. Math. 18(2), 13 pages (2016) CrossRefGoogle Scholar
  5. 5.
    Boccardo, L., de Figueiredo, D.G.: Some remarks on a system of quasilinear elliptic equations. Nonlinear Differ. Equ. Appl. 9, 309–323 (2002) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chipot, M.: Handbook of Differential Equations—Stationary Partial Differential Equations. Elsevier, Amsterdam (2008) zbMATHGoogle Scholar
  7. 7.
    de Figueiredo, D.G.: Nonlinear elliptic systems. Anais da Academia Brasileira de Ciências 72 (2000) Google Scholar
  8. 8.
    de Figueiredo, D.G., Ding, Y.H.: Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Am. Math. Soc. 355(7), 2973–2989 (2003) MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Figueiredo, D.G., Felmer, P.L.: On superquadratic elliptic systems. Trans. Am. Math. Soc. 343, 119–123 (1994) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bartsch, T., de Figueiredo, D.G.: Infinitely many solutions of nonlinear elliptic systems. Progress in Nonlinear Differential Equations and Their Applications 35 (1999) Google Scholar
  11. 11.
    Wang, J., Xu, J., Zhang, F.: Existence and multiplicity of solutions for asymptotically Hamiltonian elliptic systems in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 367, 193–203 (2010) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Clark, D.C.: A variant of the Lusternik–Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Costa, D.G.: An Invitation to Variational Methods in Differential Equations. Birkhäuser, Boston (2007) CrossRefGoogle Scholar
  14. 14.
    de Morais Filho, D.C., Souto, M.A.S.: Systems of p-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Commun. Partial Differ. Equ. 24, 1537–1553 (1999) CrossRefGoogle Scholar
  15. 15.
    Bianchi, G., Chabrowski, J., Szulkin, A.: On symmetric solutions of an elliptic equation with nonlinearity involving critical Sobolev exponent. Nonlinear Anal. 25, 41–59 (1995) MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • Patrícia Leal da Cunha
    • 1
    Email author
  • Flávio Almeida Lemos
    • 2
  1. 1.Departament of Technology and Data ScienceFundação Getulio VargasSão PauloBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Ouro PretoMinas GeraisBrazil

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