A new strict decay rate for systems of longitudinal $m$-nonlinear viscoelastic wave equations

1.   Introduction

In this paper, we are concerned with the following sub-Laplacian system with Sobolev-Hardy critical nonlinearities on Carnot group $\mathbb{G}$:

where $\Delta_{\mathbb{G}}$ stands for the sub-Laplacian operator on Carnot group $\mathbb{G}$, $\mu_{1}$, $\mu_{2}\in [0, \mu_{\mathbb{G}})$, $\alpha, \, \gamma\in (0, 2)$, $\lambda_{1}$, $\lambda_{2}$, $\beta$ are positive parameters, $p_{1}$, $p_{2} > 0$ with $1 < p_{1}+p_{2} < 2$, $\psi = |\nabla_{\mathbb{G}}\text{d}(z)|$, $\nabla_{\mathbb{G}}$ denotes the horizontal gradient and $\text{d}$ is the natural gauge associated with the fundamental solution of $-\Delta_{\mathbb{G}}$ on $\mathbb{G}$. Here, $\mu_{\mathbb{G}} = (\frac{Q-2}{2})^{2}$ is the best Hardy constant and $2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2}$ is the Sobolev-Hardy critical exponents, $Q\geq 3$ being the homogeneous dimension of the space $\mathbb{G}$ with respect to the dilation $\delta_{\gamma}$. Moreover, the function $f(z)$ satisfies the following assumption:

$(f) \; f(z)\in L^{p_{*}}(\mathbb{G}, \frac{\psi^{\gamma}}{d(z)^{\gamma}}dz)$ and the Lebesgue measure of set $\{z\in \mathbb{G}:\, f(z) > 0\}$ is positive, where $p_{*} = \frac{2^*(\gamma)}{2^*(\gamma)-(p_{1}+p_{2})}$, $0 < \gamma < 2$.

Our goal is to prove, by means of variational methods, the existence of weak solutions to (1.1). We define the energy functional $I_{\lambda_{1}, \lambda_{2}, \beta}$ associated to (1.1) as follows

defined on the product space $\mathcal{H}: = S^{1}_{0}(\mathbb{G})\times S^{1}_{0}(\mathbb{G})$, where the Folland-Stein space $S^{1}_{0}(\mathbb{G}) = \{u\in L^{2^*}(\mathbb{G}): \int_{\mathbb{G}}|\nabla_{\mathbb{G}}u|^{2}dz < +\infty\}$ is the closure of $C^{\infty}_{0}(\mathbb{G})$ with respect to the norm

Here, $2^* = \frac{2Q}{Q-2}$ is the Sobolev critical exponent. Further, we endow the product space $\mathcal{H}$ with the following norm

where

The above norm is well-defined due to the following Hardy-type inequality on Carnot group

where $\mu_{\mathbb{G}} = (\frac{Q-2}{2})^{2}$ is the optimal constant for (1.2). We can note that the norms $\|\cdot\|_{\mu_{i}}$ and $\|\cdot\|_{S^{1}_{0}(\mathbb{G})}$ for any $\mu_{i} < \mu_{\mathbb{G}}$ with $i = 1, 2$ are equivalent due to the Hardy's inequality (1.2).

The inequality (1.2) was first proved by Garofalo and Lanconelli in ^[1] for the Heisenberg group (see also ^[2]), and extended it to Carnot groups by D' Ambrosio, see ^[3]. In the Euclidean space setting, the weight function $\psi$ appearing in the l.h.s. of (1.2) is constant, i.e., $\psi\equiv1$. So, (1.2) becomes the well-known Hardy inequality:

where $\bar{\mu} = (\frac{N-2}{2})^{2}$ is the best constant and it is never attained. In the Euclidean space, the existence and non-existence, as well as qualitative properties, of nontrivial weak solutions for $p$-Laplacian equations with singular potentials and critical exponents were recently studied by several authors, we refer, e.g., in bounded domains and for $p = 2$ to ^[4,5,6,7,8], and for general $p > 1$ to ^[9,10,11,12]; while in $\mathbb{R}^n$ and for $p = 2$ to ^[13,14,15], and for general $p > 1$ to ^[16,17,18], and for fractional $(p, q)$-Laplacian to ^[19], and the references therein. Moreover, a more interesting result can be found in ^[20], which studies the critical $p$-Laplace equation on the Heisenberg group with a Hardy-type term.

In recent years, people have paid much attention to the following singular sub-elliptic problem:

where $\Omega$ is a smooth bounded domain in Carnot group $\mathbb{G}$, $0\in \Omega$. It should be mentioned that ^[21], by using Moser-type iteration, the author studied the asymptotic behavior of weak solutions to (1.3) when the function $f$ satisfies the following condition:

and obtained the following asymptotic behavior at origin:

Subsequently, in ^[22] also the behavior at infinity has been determined for the purely critical problem

for which the asymptotic estimates at the origin and at infinity are then, respectively:

where $a(\mu) = \sqrt{\mu_{\mathbb{G}}}-\sqrt{\mu_{\mathbb{G}}-\mu}$, $b(\mu) = \sqrt{\mu_{\mathbb{G}}}+\sqrt{\mu_{\mathbb{G}}-\mu}$ and the notation $f\sim g$ means that there exists a constant $C > 0$ such that $\frac{1}{C}g(z)\leq f(z)\leq C g(z)$. From a technical point of view, these asymptotic estimates have a fundamental role in the study of the associated Brezis-Nirenberg type sub-elliptic problems on Carnot group. For more details on this topic, please refer to ^[22], which provides a detailed analysis of the Brezis-Nirenberg problem on Carnot group.

Motivated by the aforementioned articles and their results, we are interested in finding existence and multiplicity results for a system with critical Sobolev-Hardy critical terms. While dealing with the system (1.1), if we suppose $\mu_{1} = \mu_{2} = \mu$, $\lambda_{1} = \lambda_{2} = 1$ and $\beta = 0$, problem (1.1) reduces to a sub-elliptic critical problem

In 2015, Loiudice in the paper ^[23] proved the existence of ground state solutions of (1.4) using variational approach for $\mu = 0$ and $0 < \alpha < 2$, and obtained the asymptotic behavior of this solution at infinity. Recently, Zhang ^[24] proved the existence of ground state solutions of (1.4) $0 < \mu < \mu_{\mathbb{G}}$ and $0 < \alpha < 2$ and considered the following sub-elliptic system with critical Sobolev-Hardy nonlinearities on Carnot group

where $\alpha\in (0, 2)$, $\lambda > 0$ and $\eta$, $\theta > 1$. The existence of nontrivial solutions of the above sub-Laplacian system through variational methods was obtained for the critical case, i.e., $\eta+\theta = 2^*(\alpha)$. Other subelliptic problems with multiple critical exponents can be found in ^[25] and the references therein.

Let us recall that solutions of (1.4) arise as minimizers $u\in S^{1}_{0}(\mathbb{G})$ of the following Rayleigh quotient:

Actually, up to a normalization, it holds that

Moreover, for any $\varepsilon > 0$, rescaled functions $u_{\varepsilon}(z) = \varepsilon^{-\frac{Q-2}{2}}u(\delta_{\frac{1}{\varepsilon}}(z))$ are solutions, up to multiplicative constants, of the equation (1.4) and satisfy (1.5) too. However, the explicit form of ground state solutions is unknown, which is also the focus of our future work.

As a natural extension of the above papers, we are mainly interested in searching infinitely many solutions of singular sub-elliptic problem (1.1). Our point is here a combination of sub-Laplace operator and critical Sobolev-Hardy terms on the Carnot group. In the Euclidean elliptic setting, i.e., when $\mathbb{G}$ is the ordinary Euclidean space $(\mathbb{R}^N, +)$, starting with the pioneering work of Kajikiya ^[26], established a critical point theorem related to the symmetric mountain pass lemma and applied it to find the existence of infinitely many solutions to elliptic equation. A large number of scholars have investigated the application of this method and achieved rich results, such as He-Zou ^[27], Baldelli-Filippucci ^[28], Liang-Zhang ^[29,30], Ambrosio-Isernia ^[19] and Liang-Shi ^[31] in this direction.

Motivated by the above results, our aim of this paper is to show the existence of infinitely many solutions of sub-elliptic problem (1.1), and that there exists a sequence of infinitely many arbitrarily small solutions converging to zero using the symmetric mountain-pass lemma due to Kajikiya ^[26]. To the best of our knowledge, there are only some results that deal with the sub-Laplacian problem with Sobolev-Hardy critical exponents and Hardy-type terms on the Carnot group.

Before stating our main result, let us recall the definition of weak solutions to (1.1).

Definition 1.1. We say that $(u, v)\in \mathcal{H}$ is a weak solutions of (1.1), if $(u, v)$ satisfies

for all $(\phi_{1}, \phi_{2})\in \mathcal{H}$.

By Hardy-Sobolev inequality, it is clear that $I_{\lambda_{1}, \lambda_{2}, \beta}$ is well-defined on $\mathcal{H}$ and belongs to $C^{1}(\mathcal{H}, \mathbb{R})$. Then, from Definition 1.1 we see that any weak solution of (1.1) is just a critical point of $I_{\lambda_{1}, \lambda_{2}, \beta}$. Therefore, we are now in position to state our main result as follows.

Theorem 1.1. Assume that $(f)$ holds, and $1 < p_{1}+p_{2} < 2$, $0\leq \alpha < 2$, $0\leq \gamma < 2$. Then

(i) for any $\beta > 0$, there exists $\widetilde{\lambda} > 0$ such that if $0 < \lambda_{1} < \widetilde{\lambda}$, $0 < \lambda_{2} < \widetilde{\lambda}$, problem (1.1) has a sequence of solutions $\{(u_n, v_{n})\}\subset \mathcal{H}$ with $I_{\lambda_{1}, \lambda_{2}, \beta}(u_n, v_{n}) < 0$ and $I_{\lambda_{1}, \lambda_{2}, \beta}(u_n, v_{n})\to 0$ as $n\to \infty$.

(ii) for any $\lambda_{1}$, $\lambda_{2} > 0$, there exists $\widetilde{\beta} > 0$ such that if $0 < \beta < \widetilde{\beta}$, problem (1.1) has a sequence of solutions $\{(u_n, v_{n})\}\subset \mathcal{H}$ with $I_{\lambda_{1}, \lambda_{2}, \beta}(u_n, v_{n}) < 0$ and $I_{\lambda_{1}, \lambda_{2}, \beta}(u_n, v_{n}) \to 0$ as $n\to \infty$.

Remark 1.1 Using the symmetric mountain pass lemma (see Theorem 2.1) we can conclude that the solutions obtained from Theorem 1.1 satisfy $(u_n, v_{n}) \to (0, 0)$ as $n\to \infty$.

The main idea to prove Theorem 1.1 is based on concentration-compactness result on the Carnot group and the symmetric mountain pass lemma ^[26]. One of the main difficulties to prove the existence and multiplicity of solutions of equation (1.1) using variational methods is that the energy functional does not satisfy the Palais-Smale condition for large energy levels, since the embedding $S^{1}_{0}(\mathbb{G})\hookrightarrow L^{2^*(\alpha)}(\mathbb{G}, \frac{\psi^{\alpha}}{\text{d}(z)^{\alpha}}dz)$ is not compact. Another difficulty is that every nontrivial solution of (1.1) is singular at $\{z = 0\}$ due to the presence of the Hardy terms. Thus, different techniques are needed to deal with the singular case.

The rest of this paper is organized as follows. In Section 2, the variational setting and some preliminary are recalled. Finally, Section 3 contains several preliminary lemmas, including the crucial concentration-compactness lemma, as well as the proof of Theorem 1.1.

2.   Preliminary

We devote this section to state some useful facts on the Carnot groups. For more details, we refer the reader to ^{[32,33,34,35,36]} and references therein.

A Carnot group (or Stratified group) $(\mathbb{G}, \circ)$ is a connected, simply connected nilpotent Lie group, whose Lie algebra $\mathfrak{g}$ admits a stratification, namely a decomposition $\mathfrak{g} = \oplus_{k = 1}^{r}V_k$ with

Here, the integer $r$ is called the step of $\mathbb{G}$, $\dim(V_k) = N_k$ and the symbol $[V_1, V_k]$ denotes the subspace of $\mathfrak{g}$ generated by the commutators $[X, Y]$, where $X\in V_1$ and $Y\in V_k$.

By means of the natural identification of $\mathbb{G}$ with its Lie algebra via the exponential map, it is not restrictive to suppose that $\mathbb{G}$ is a homogeneous group, i.e., Lie group equipped with a family $\{\delta_{\gamma}\}_{\gamma > 0}$ of dilations, acting on $z\in \mathbb{R}^{N}$ as follows

where $z^{(k)}\in \mathbb{R}^{N_k}$ for every $k\in \{1, \cdots, r\}$ and $N = \sum_{k = 1}^{r}N_k$. Then, the structure $\mathbb{G}: = (\mathbb{R}^N, \circ, \{\delta_{\gamma}\}_{\gamma > 0})$ is called a homogeneous group with homogeneous dimension $Q: = \sum_{k = 1}^{r}k\cdot N_k$. Note that the number $Q$ is naturally associated to the family $\{\delta_{\gamma}\}_{\gamma > 0}$ since, for every $\gamma > 0$, the Jacobian of the map $z\mapsto \delta_{\gamma}(z)$ equals $\gamma^{Q}$. Moreover, the number $N: = \sum_{k = 1}^{r}N_k$ is called the topological dimension of $\mathbb{G}$.

Now, let $\{X_1, \cdots, X_{N_1}\}$ be any basis of $V_1$, the sub-Laplacian on $\mathbb{G}$ is define as the second order differential operator

The horizontal gradient on $\mathbb{G}$ is define as

The horizontal divergence on $\mathbb{G}$ is define by

It is easy to check that $\nabla_\mathbb{G}$ and $\Delta_{\mathbb{G}}$ are left-translation invariant with respect to the group action $\tau_{z}$ and $\delta_{\gamma}$-homogeneous, respectively, of degree one and two, that is, $\nabla_\mathbb{G}(u\circ \tau_{z}) = \nabla_\mathbb{G}u\circ \tau_{z}$, $\nabla_\mathbb{G}(u\circ \delta_{\gamma}) = \gamma \nabla_\mathbb{G}u\circ \delta_{\gamma}$; $\Delta_\mathbb{G}(u\circ \tau_{z}) = \Delta_\mathbb{G}u\circ \tau_{z}$ and $\Delta_\mathbb{G}(u\circ \delta_{\gamma}) = \gamma^{2} \Delta_\mathbb{G}u\circ \delta_{\gamma}$, where the left translation $\tau_{z}:\mathbb{G}\to \mathbb{G}$ is defined by

Let us now define the homogeneous norm Carnot group $\mathbb{G}$.

Definition 2.1 A continuous function $\text{d}:\mathbb{G}\to [0, +\infty)$ is said to be a homogeneous norm on $\mathbb{G}$ if it satisfies the following condition:

(i) $\text{d}(z) = 0$ if and only if $z = 0$;

(ii) $\text{d}(z^{-1}) = \text{d}(z)$ for all $z\in \mathbb{G}$;

(iii) $\text{d}(\delta_{\gamma}(z)) = \gamma \text{d}(z)$ for every $\gamma > 0$ and $z\in \mathbb{G}$.

Throughout this paper, we almost exclusively work with the homogeneous norm, which is related to the fundamental solution of the sub-Laplace operator $-\Delta_{\mathbb{G}}$, that is the function $\text{d}$ such that

is the fundamental solution of $-\Delta_{\mathbb{G}}$ with pole at $0$, for a suitable constant $C > 0$, see ^[22,33]. Moreover, if we define $\text{d}(z_{1}, z_{2}): = \text{d}(z_{2}^{-1}\circ z_{1})$, then $\text{d}$ is a pseudo-distence on $\mathbb{G}$. In particular, $\text{d}$ satisfies the pseudo-triangular inequality:

for a suitable positive constant $c$. The ball of radius $R > 0$ centered at $z\in \mathbb{G}$ with respect to the norm $\text{d}$, calling them $\text{d}$-balls, defined as

In fact, the norm on $\mathbb{G}$ can be induced by the Euclidean distance $|\cdot|$ on $\mathfrak{g}$ through the exponential mapping, which also induces the homogeneous pseudo-norm $|\cdot|_{\mathfrak{g}}$ on $\mathfrak{g}$, namely, for $\xi \in \mathfrak{g}$ with $\xi = \xi_{1}+\cdots +\xi_{k}$, where $\xi_{i}\in V_{i}$, define a pseudo-norm on $\mathfrak{g}$ as follows

The induced norm on $\mathbb{G}$ has the form

The function $|\cdot|_{\mathbb{G}}$ is usually known as the non-isotropic gauge. It defines a pseudo-distence on $\mathbb{G}$ given by

The simplest example of a stratified Lie group is the Heisenberg group $\mathbb{H}^{N}: = (\mathbb{R}^{2N+1}, \circ)$ with the composition law as

where $(x, y, t), (x', y, 't')\in \mathbb{R}^{N}\times \mathbb{R}^{N}\times \mathbb{R}^{1}$ and $\langle \cdot, \cdot\rangle$ represents the inner product on $\mathbb{R}^{N}$. The sub-Laplacian on $\mathbb{H}^{N}$ is given by

where

In order to prove Theorem 1.1, we will recall some basic facts involved in the so-called Krasnoselskii genus, which can be found in ^[37,38].

For a symmetric group $\mathbb{Z}_{2} = \{\text{id}, \, -\text{id}\}$ and let $E$ be a Banach space we set

For any $A\in\Sigma$, the Krasnoselskii's genus of $A$ is defined by

If $k$ does not exist, we set $\gamma(A) = \infty$. By above definition, it is obvious that $\gamma(\emptyset) = 0$.

Let $\Sigma_{k}$ denote the family of closed symmetric subsets $A$ of $E$ such that $0\not\in E$ and $\gamma(A)\geq k$, that is,

Then we have the following result, see ^[26,37].

Proposition 2.1. Let $A$ and $B$ be closed symmetric subsets of $E$ which do not contain the origin. Then the following statements hold:

(1) If there exists an odd continuous mapping from $A$ to $B$, then $\gamma(A) \leq \gamma(B)$.

(2) If $A\subset B$, then $\gamma(A) \leq \gamma(B)$.

(3) If there is an odd homeomorphism from $A$ to $B$, then $\gamma(A) = \gamma(B)$.

(4) If $\gamma(B) < \infty$, then $\gamma\overline{(A\setminus B)} \geq \gamma(A) - \gamma(B)$.

(5) If $S^n$ is a $n$-dimensional sphere, then $\gamma(S^n) = n+1$.

(6) If $A$ is compact, then $\gamma (A) < +\infty$ and there exists a $\delta$-closed symmetric neighborhood of $A$, i.e., $N_\delta(A) = \left\{u\in E: \operatorname{dist}(u, A)\leq\delta\right\}$ such that $N_{\delta} (A)\subset\Sigma_{k}$ and $\gamma (N_{\delta} (A)) = \gamma (A).$

Now, we state the following variant of symmetric mountain-pass lemma due to Kajikiya ^[26].

Theorem 2.1. Let $E$ be an infinite-dimensional Banach space, and let $J \in C^1(E, \mathbb{R})$ be a functional satisfying the conditions below:

(1) $J(u)$ is even, bounded from below, $J(0) = 0$ and $J(u)$ satisfies the local Palais-Smale condition, i.e. for some $\bar{c} > 0$, every sequence $\{u_n\}$ in $E$ satisfying $\lim\limits_{n\to\infty}J(u_n) = c < \bar{c}$ and $\lim\limits_{n\to\infty}\|J'(u_n)\|_{E'} = 0$ has a convergent subsequence;

(2) For each $k \in \mathbb{N}$, there exists $A_k \in \Sigma_k$ such that $\sup\limits_{u\in A_k}J(u) < 0$.

Then either (i) or (ii) below holds.

(i) There exists a sequence $\{u_n\}$ such that $J'(u_n) = 0$, $J(u_n) < 0$ and $\{u_n\}$ converges to zero as $n\to \infty$.

(ii) There exist two sequences $\{u_n\}$ and $\{v_n\}$ such that $J'(u_n) = 0$, $J(u_n) = 0$, $u_n \neq 0$, $\lim\limits_{n \to \infty}u_n = 0$; $J'(v_n) = 0$, $J(v_n) < 0, \; \lim\limits_{n \to \infty}J(v_n) = 0$, and $\{v_n\}$ converges to a non-zero limit.

3.   Proof of Theorem 1.1

In this section, we first discuss a compactness property for the energy functional $I_{\lambda_{1}, \lambda_{2}, \beta}$, given by the Palais-Smale condition.

Let $c\in \mathbb{R}$, $\mathcal{H}$ be a Banach space and $I_{\lambda_{1}, \lambda_{2}, \beta}\in C^{1}(\mathcal{H}, \mathbb{R})$. $\{(u_{n}, v_{n})\}\subset \mathcal{H}$ is a Palais-Smale sequence for $I_{\lambda_{1}, \lambda_{2}, \beta}$ in $\mathcal{H}$ at level $c$, $(PS)_c$-sequence for short, if

We say that $I_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies $(PS)_{c}$-condition at level $c$ if for any $(PS)_{c}$-sequence $\{(u_n, v_{n})\}\subset \mathcal{H}$ for $I_{\lambda_{1}, \lambda_{2}, \beta}$ has a convergent subsequence in $\mathcal{H}$.

In order to apply Theorem 2.1, we need the following preliminary results for $(PS)_{c}$-sequence of $I_{\lambda_{1}, \lambda_{2}, \beta}$.

Lemma 3.1. Suppose that $1 < p: = p_{1}+p_{2} < 2$ and $\alpha, \, \gamma\in (0, 2)$. Let $\{(u_{n}, v_{n})\}\subset \mathcal{H}$ be a $(PS)_{c}$-sequence for $I_{\lambda_{1}, \lambda_{2}, \beta}$. Then, $\{(u_{n}, v_{n})\}$ is bounded in $\mathcal{H}$.

Proof. Let $\{(u_{n}, v_{n})\}\subset \mathcal{H}$ be a $(PS)_{c}$-sequence for $I_{\lambda_{1}, \lambda_{2}, \beta}$, then

By Young inequality and Hölder inequality, we have

Then,

which implies that $\{(u_{n}, v_{n})\}$ is bounded in $\mathcal{H}$ since $p < 2 < 2^*(\alpha)$ and $\beta > 0$.

Proposition 3.1. Let $1 < p < 2$, $\alpha, \, \gamma\in (0, 2)$ and let $\{(u_{n}, v_{n})\}\subset \mathcal{H}$ be a $(PS)_{c}$-sequence of $I_{\lambda_{1}, \lambda_{2}, \beta}$ with $c < 0$. Then,

(i) for any $\lambda_{1}$, $\lambda_{2} > 0$, there exists $\beta_{*} > 0$ such that if $0 < \beta < \beta_{*}$, $I_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies $(PS)_{c}$ condition, where $\beta_{*}$ is independent on the sequence $\{(u_{n}, v_{n})\}$;

(ii) for any $\beta > 0$, there exists $\lambda_{*} > 0$ such that is $0 < \lambda_{1} < \lambda_{*}$, $0 < \lambda_{2} < \lambda_{*}$, $I_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies $(PS)_{c}$ condition, where $\lambda_{*}$ is independent on the sequence $\{(u_{n}, v_{n})\}$.

Proof. Since the sequence $\{(u_{n}, v_{n})\}$ is bounded in $\mathcal{H}$, thanks to Lemma 3.1, then there exists $(u_{0}, v_{0})\in \mathcal{H}$ such that, up to a subsequence, it follows that

Then, by the concentration-compactness principle ^[39,40,41] and up to a subsequence if necessary, there exist positive finite Radon measure $\hat{\mu}$, $\hat{\nu}$, $\hat{\rho}$, $\bar{\mu}$, $\bar{\nu}$, $\bar{\rho}\in \mathcal{R}(\mathbb{G}\cup \{\infty\})$; at most countable set $J$ and $\bar{J}$; real numbers $\hat{\mu}_{j}$, $\hat{\nu}_{j} \; (j\in J)$, $\bar{\mu}_{k}$, $\bar{\nu}_{k}\, (k\in \bar{J})$, $\hat{\mu}_{0}$, $\hat{\nu}_{0}$, $\hat{\rho}_{0}$, $\bar{\mu}_{0}$, $\bar{\nu}_{0}$, $\bar{\rho}_{0}$ and different points $z_{j}\in \mathbb{G}\backslash\{0\}$ ($j\in J$), $\bar{z}_{k}\in \mathbb{G}\backslash\{0\}$ ($k\in \bar{J}$) such that

where $\delta_{z}$ is the Dirac mass at $z$. Moreover, by the Sobolev-Hardy and the Hardy inequalities, we get

where $S(\alpha, \mathbb{G})$ is the best Hardy-Sobolev constant, i.e.,

In order to study the concentration at infinity of $\{u_{n}\}$ and $\{v_{n}\}$, we use a method of concentration-compactness principle at infinity, which was first established by Chabrowski ^[42]. We set

and

For the sequence $\{u_{n}\}$, let $\phi_{j}(z) \in C^{\infty}_{0}(\mathbb{G}, [0, 1])$ be a cut-off function centered at $z_{j}\in \mathbb{G}\backslash\{0\}$ with $\phi_{j} = 1$ on $B_{d}(z_{j}, 1)$, $\phi_{j} = 0$ on $\mathbb{G}\backslash B_{d}(z_{j}, 2)$. Let $\phi_{j, \varepsilon}(z) = \phi_{j}(\delta_{\frac{1}{\varepsilon}}(z))$. Then $|\nabla_{\mathbb{G}} \phi_{j, \varepsilon}|\leq \frac{C}{\varepsilon}$ and $\{\phi_{j, \varepsilon}\, u_{n}\}$ is bounded in $S^{1}_{0}(\mathbb{G})$. Testing $I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})$ with $(\phi_{j, \varepsilon}u_{n}, 0)$, we obtain $\lim_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}), (\phi_{j, \varepsilon} u_{n}, 0)\rangle = 0$, that is,

Now, we estimate each term in (3.12). From (3.1)–(3.6), we get

and

From Hölder inequality, it follows that

Consequently, from the above arguments (3.13)–(3.17), we get

Combining with (3.7), we have

which implies that the set $J$ is finite.

Similarly, for $\bar{\nu}_{k}$ and $\bar{J}$, the following conclusion holds:

On the other hand, choosing a suitable cutoff function centered at the origin, by the analogous argument we can prove that

It follows from the definition of $S_{\alpha, \mu_{1}}$ and $S_{\alpha, \mu_{2}}$ that

Thus, by combining (3.18) and (3.19), (3.20) we get

and

Furthermore, the Hardy inequality (1.2) implies that

and

If $\hat{\nu}_{0} = 0$, from (3.18) and (3.23), it follows that $\hat{\mu}_{0} = \hat{\rho}_{0} = 0$. Similarly, if $\bar{\nu}_{0} = 0$, by (3.18) and (3.24), we conclude $\bar{\mu}_{0} = \bar{\rho}_{0} = 0$.

To analyze the concentration at infinity, for $R > 0$, we choose the function $\phi\in C^{\infty}_{1}(\mathbb{G})$ such that $0\leq \phi\leq 1$, $\phi(z) = 0$ on $B_{d}(0, 1)$, $\phi(z) = 1$ on $\mathbb{G}\backslash B_{d}(0, 2)$ and $|\nabla_{\mathbb{G}}\phi|\leq \frac{c}{R}$. Set $\phi_{R}(z) = \phi(\delta_{\frac{1}{R}}(z))$, then $\{\phi_{R}u_{n}\}\subset S^{1}_{0}(\mathbb{G})$ is bounded. Testing $I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})$ with $(\phi_{R}u_{n}, 0)$ we obtain $\lim\limits_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})\, ,\, (\phi_{R}u_{n}, 0)\rangle = 0,$ i.e.,

Since

we conclude that

By Hölder inequality, it is easy to get that

Similarly,

Thus, we see from(3.27), (3.28) and (3.26), we have

On the other hand, from Hölder inequality and the definition of $\phi_{R}$ we have

Since $f\in L^{p_{*}}(\mathbb{G}, \frac{\psi^{\gamma}}{d(z)^{\gamma}}dz)$, it follows that

Thus, taking limits by letting $n\to \infty$ in (3.25), we have

Hence, it follows from (3.29) and (3.30) that

In contrast, the Hardy inequality implies that

If $\nu_{\infty} = 0$, by combining (3.30) and (3.31), we get $\mu_{\infty} = \rho_{\infty} = 0$.

From above argument the same conclusion holds for $\bar{\nu}_{\infty}$, namely,

and

If $\bar{\nu}_{\infty} = 0$, we have that $\bar{\mu}_{\infty} = \bar{\rho}_{\infty} = 0$.

Now we claim that $(2)$, $(2)'$, $(4)$, $(4)'$ and $(6)$, $(6)'$ cannot occur if $\lambda_{1}$, $\lambda_{2}$ and $\beta$ are chosen properly. In fact, applying $(f)$ and Hölder inequality, we have

Since

which and (3.32) yield that

namely,

If $(6)$ or $(6)'$ occurs, we obtain by (3.32) and (3.33) that

that is,

From the above inequality, we can find that if $\beta > 0$ is given, there exists $\lambda_{*} > 0$ small enough such that for $\lambda_{1}, \, \lambda_{2}\in (0, \lambda_{*})$, the right-hand side of (3.34) is greater than $0$, which is a contradiction. Similarly, if $\lambda_{1}, \, \lambda_{2} > 0$ is given, we can take $\beta_{*} > 0$ so small that for $\beta\in (0, \beta_{*})$, right-hand side of (3.34) is greater than $0$.

Similarly we can prove that $(2)$, $(2)'$ and $(4)$, $(4)'$ cannot occur. So

and

In view of $(u_{n}, v_{n})\rightharpoonup (u_{0}, v_{0 })$ weakly in $\mathcal{H}$ and the Brezis-Lieb lemma ^[38], we have

We are now going to prove that $(u_{n}, v_{n})\to (u_{0}, v_{0})$ strongly in $\mathcal{H}$. First, we have

For the first term in (3.35), by using Hölder inequality, we get that

Similarly,

On the other hand, using the Hölder inequality and $(u_{n}, v_{n})\rightharpoonup (u_{0}, v_{0})$ weakly in $\mathcal{H}$, we get that

and

Combining (3.36), (3.37), (3.38), (3.39), (3.35) with $\lim\limits_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}), (u_{n}-u_{0}, v_{n}-v_{0})\rangle = 0$ and $\lim\limits_{n\to \infty}\langle I'_{\lambda_{1}, \lambda_{2}, \beta}(u_{0}, v_{0}), (u_{n}-u_{0}, v_{n}-v_{0})\rangle = 0$, we deduce that

The proof is completed.

In the end of this section, we will prove the existence of infinitely many weak solutions of (1.1) which tend to zero. First, by using Hölder's inequality and Young's inequality, we get

and

Then,

Define the function

where

Because $1 < p < 2 < 2^*(\alpha)$, for the given $\beta > 0$, there exists $\lambda_{**} > 0$ so small that for $\lambda_{1}+\lambda_{2} \in (0, \lambda_{**})$, there exist $t_{1}$, $t_{2} > 0$ with $t_1 < t_2$ such that $g(t_1) = g(t_2) = 0$, and $g(t) < 0$ for $t \in (0, t_1)$, $g(t) > 0$ for $t \in (t_{1}, t_2)$, $g(t) < 0$ for $t \in (t_{2}, +\infty)$. Similarly, given $\lambda_{1}$, $\lambda_{2} > 0$, we can choose $\beta_{**} > 0$ small enough such that for all $\beta \in(0, \beta_{**})$, there exist $\hat{t}_{1}$, $\hat{t}_{2} > 0$ with $\hat{t}_1 < \hat{t}_2$ such that $g(\hat{t}_1) = g(\hat{t}_2) = 0$ and $g(t) < 0$ for $t\in (0, \hat{t}_1)$, $g(t) > 0$ for $t\in (\hat{t}_1, \hat{t}_2)$, $g(t) < 0$ for $t\in (\hat{t}_{2}, +\infty)$.

Let us define a function $\phi\in C^{\infty}_{0}([0\, ,\, \infty), \mathbb{R})$ such that $0\leq \phi(t)\leq 1$, $\phi(-t) = \phi(t)$ for all $t\in [0, +\infty)$, $\phi(t) = 1$ if $t\in [0, t_{1}]$ and $\phi(t) = 0$ if $t\in [t_{2}, \infty)$. So we consider the equation

and we observe that if $(u, v)$ is a weak solution of (3.41) such that $\|(u, v)\|_{\mathcal{H}} < t_{1}$, then $(u, v)$ is also a solution of (1.1). For this reason we look for critical points of the following functional $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}: \mathcal{H}\to \mathbb{R}$ defined as

In view of the definition of $\phi$ and $p < 2$ we can see that $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)\to \infty$ as $\|(u, v)\|_{\mathcal{H}}\to \infty$, $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(-u, -v) = \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)$ and $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)$ is bounded from below. Moreover, $I_{\lambda_{1}, \lambda_{2}, \beta}(u, v)\leq \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)$ for all $(u, v)\in \mathcal{H}$.

Next, we show that $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies the assumptions of Theorem 2.1.

Lemma 3.2. (i) If $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) < 0$, then $\|(u, v)\|_{\mathcal{H}} < t_1$ and $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v}) = I_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v})$ for all $(\tilde{u}, \tilde{v})\in N_{(u, v)}$, where $N_{(u, v)}$ denotes the enough neighborhood of $(u, v)$.

(ii) For $\lambda_{1}$, $\lambda_{2} > 0$, there exists $\widetilde{\beta } = \min\{\beta_{*}, \beta_{**}\}$ such that if $\beta \in (0, \widetilde{\beta})$ and $c \in (-\infty, 0)$, then $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies $(PS)_c$-condition;

(iii) For $\beta > 0$, there exists $\widetilde{\lambda} = \min\{\lambda_{*}, \lambda_{**}\}$ such that if $\lambda_{1}, \, \lambda_{2} \in(0, \widetilde{\lambda})$ and $c \in(-\infty, 0)$, then $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies $(PS)_c$-condition.

Proof. We prove (i) by contradiction, assume $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u), v\leq0$ and $\|(u, v)\|_{\mathcal{H}}\geq t_1$. If $\|(u, v)\|_{\mathcal{H}}\geq t_2$, then we have

This contradicts $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) < 0$.

If $t_{1}\leq \|(u, v)\|_{\mathcal{H}} < t_2$, since $0\leq\phi(t)\leq 1$, we get

which again contradicts $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) < 0$. Hence, $\|(u, v)\|_{\mathcal{H}} < t_1$. Furthermore, by continuity of $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$, applying $I_{\lambda_{1}, \lambda_{2}, \beta}(u, v) = \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)$ for all $\|(u, v)\|_{\mathcal{H}} < t_1$ there exists a small neighborhood $\mathcal{B}_{(u, v)}\subset B_{\text{d}}((0, 0), R)$ of $(u, v)$ such that $I_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v}) = \mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(\tilde{u}, \tilde{v})$ for any $(\tilde{u}, \tilde{v})\in \mathcal{B}_{(u, v)}$, we conclude the proof of (i).

Now we prove (ii), let $\widetilde{\beta} = \min\{\beta_{*}, \beta_{**}\}$, and let $\{(u_{n}, v_{n})\}\subset \mathcal{H}$ be a $(PS)_{c}$-sequence for $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ with the level $c < 0$, then $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})\to c$ and $\mathcal{J}'_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})\to 0$ in $\mathcal{H}^{-1}$. By (i), we have $\|(u_{n}, v_{n})\|_{\mathcal{H}} < t_1$, hence $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n}) = I_{\lambda_{1}, \lambda_{2}, \beta}(u_{n}, v_{n})$. By Proposition 3.1, $I_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies the $(PS)_{c}$-condition for $c < 0$. Thus, $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies the $(PS)_{c}$-condition for $c < 0$, (ii) holds.

The proof of (iii) goes exactly as (ii) with only minor modification, we omit it here.

Let

Lemma 3.3. Given $k\in \mathbb{N}$, there exists $\varepsilon = \varepsilon(k) > 0$ such that $\gamma(\mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta})\geq k$ for any $\lambda_{1}$, $\lambda_{2}$, $\beta > 0$.

Proof. Fix $\lambda_{1}$, $\lambda_{2} > 0$, $k\in \mathbb{N}$ and let $E_{k}$ be a $k$-dimensional vectorial subspace of $\mathcal{H}$. Taking $(u, v)\in E_{k}\backslash\{(0, 0)\}$ with $(u, v) = r_{k}(\omega_{1}, \omega_{2})$, where $(\omega_{1}, \omega_{2})\in E_{k}$ and $\|(\omega_{1}, \omega_{2})\|_{\mathcal{H}} = 1$. Then, by (3.40) there is a constant $C > 0$ such that

which implies that there exists $c_{k}\in (-\infty, +\infty)$ such that

Thus, for each $(u, v) = r_{k}(\omega_{1}, \omega_{2})$ with $r_{k}\in (0, t_{1})$, we have

For any $\varepsilon: = \varepsilon(k) > 0$, there exists $r_{k}\in (0, t_{1})$ small enough such that $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v)\leq -\varepsilon$ for any $(u, v)\in \mathcal{H}$ with $\|(u, v)\|_{\mathcal{H}} = r_{k}$.

Denote $\mathcal{S}_{k} = \{(u, v)\in \mathcal{H}:\, \|(u, v)\|_{\mathcal{H}} = r_{k}\}.$ Clearly, $\mathcal{S}_{k}$ is homeomorphic to the $k-1$ dimensional sphere $\mathbb{S}^{k-1}$ and $\mathcal{S}_{k}\cap E_{k}\subset \mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta}$. By Proposition 2.1 (2) and (4) it follows that

concluding the proof.

Let us set the number

with

Clearly, $c_{k}\leq c_{k+1}$ for each $k\in \mathbb{N}$. Before proving our main result, we state the following technical results.

Lemma 3.4. $c_{k} < 0$ for all $k\in \mathbb{N}$.

Proof. Fix $k\in \mathbb{N}$. By Lemma 3.3, there exists $\varepsilon > 0$ such that $\gamma(\mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta})\geq k$. This and $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ is a continuous even functional imply that $\mathcal{J}^{-\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta}\in \Gamma_{k}$. Then

Therefore, taking into account that $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ is bounded from below, we get

Let

Lemma 3.5. For any $\lambda_{1}$, $\lambda_{2}$, $\beta > 0$, the critical values $\{c_k\}_{k\in \mathbb{N}}$ of $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfy $c_{k}\to 0$ as $k\to \infty$.

Proof. Fix $\mu_{1}$, $\mu_{2}\in [0, \mu_{\mathbb{G}})$ and $\lambda_{1}$, $\lambda_{2}$, $\beta > 0$. By Lemma 3.4 it follows that $c_{k} < 0$. Since $c_{k}\leq c_{k+1}$ we can assume that $\lim\limits_{k\to \infty}c_{k}\to c_{0} \leq 0$. Moreover, by Lemma 3.2, it is easy to see that the functional $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies the $(PS)_{c_{k}}$-condition at level $c_{k}$.

Now we prove that $c_{0} = 0$. We argue by contradiction and we suppose that $c_{0} < 0$. In view of Lemma 3.2, $K_{c_{0}}$ is compact. Furthermore, it is easy to see that

which and Proposition 2.1 (6) imply that $\gamma(K_{c_{0}}) = k_{0} < \infty$ and there exists $\delta > 0$ such that $N_{\delta}(K_{c_0})\subset \mathcal{E}$ and

where $N_{\delta}(K_{c_{0}}) = \{(u, v)\in \mathcal{H}:\text{dist}((u, v), K_{c_0})\leq \delta\}$. Moreover, By [38, Theorem A.4], there exists an odd homeomorphism $\eta:\, \mathcal{H}\to \mathcal{H}$ such that

Taking into account that $c_{k+1}\leq c_{k}$ and $c_{k}\to c_{0}$ as $k\to \infty$, we can find $k\in \mathbb{N}$ such that $c_{k} > c_{0}-\varepsilon$ and $c_{k+k_{0}}\leq c_{0}$, where $k_{0}$ given in (3.42). Take $A \in \Gamma_{k+k_{0}}$ such that $\sup\limits_{(u, v)\in A}\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}(u, v) \leq c_{k+k_{0}} < c_{0} +\varepsilon$, by using Properties 2.1 (4), we have

from which we have $\eta(\overline{A\setminus N_{\delta}(K_{c_{0}})})\in \Gamma_{k}$. Hence

On the other hand, in view of (3.43) and $A\subset \mathcal{J}^{c_{0}+\varepsilon}_{\lambda_{1}, \lambda_{2}, \beta}$, we see that

which gives a contradiction in virtue of (3.44). Hence, $c_{0} = 0$ and $\lim\limits_{k\to \infty}c_{k} = 0$ hold.

Lemma 3.6. Let $\lambda_{1}$, $\lambda_{2}$, $\beta$ be as in (ii) or (iii) of Lemma 3.2. If $k, \, l\in\mathbb{N}$ such that $c = c_{k} = c_{k+1} = \cdots = c_{k+l}$, then

Proof. From Lemma 3.4 we have that $c = c_k = c_{k+1} = \ldots = c_{k+l} < 0$. By Lemma 3.2, $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies the $(PS)_{c}$-condition on the compact set $K_{c}$.

Suppose the result is not true, that is, $\gamma(K_c)\leq l$. Then, by Proposition 2.1 (6) there is a neighborhood of $K_{c}$, say $N_\delta(K_c)$, such that $\gamma(N_\delta(K_c)) = \gamma(K_c)\leq l$. By [38, Theorem A.4], there exists an odd homeomorphism $\eta: \mathcal{H}\to \mathcal{H}$ such that

From the definition of $c = c_{n+l}$, we know there exists $A\in\Gamma_{n+l}$ such that

that is, $A\subset\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}^{c+\varepsilon}$, and so by (3.45) we get

This yields

On the other hand, by parts (1), (3) of Proposition 2.1 we have

Hence, we conclude that $\eta(\overline{A \setminus N_\delta(K_{c})}) \in \Gamma_n$ and so

which contradicts (3.46). Thus, we conclude $\gamma(K_{c})\geq l+1$.

Proof of Theorem 1.1 Let $\lambda_{1}$, $\lambda_{2}$, $\beta$ be as in (ii) or (iii) of Lemma 3.2. Putting together Lemma 3.4 and Lemma 3.2 (ii) or (iii), we can see that the functional $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ satisfies the $(PS)_{c_{k}}$-condition with $c_n < 0$. That is, $c_k$ is a critical value of $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$.

We consider two situations.

If all $c_{k}$'s are distinct, that is, $-\infty < c_1 < c_2 < \cdots < c_k < c_{k+1} < \cdots$, then $\gamma(K_{c_{k}})\geq 1$ since $K_{c_{k}}$ is a compact set. Thus, in this case $\mathcal{J}_{\lambda_{1}, \lambda_{2}, \beta}$ admits infinitely many critical values. By Lemma 3.2 (1) we can see that $I_{\lambda_{1}, \lambda_{2}, \beta}$ has infinitely many critical points, i.e., (1.1) has infinitely many solutions.

If for some $k\in \mathbb{N}$ there exists $l\in\mathbb{N}$ such that $c_k = c_{k+1} = \cdots = c_{k+l} = c$, then $\gamma(K_c)\geq l+1\geq2$ by Lemma 3.6. Thus, the set $K_c$ has infinitely many distinct elements, (see [38, Remark 7.3]), i.e., $I_{\lambda_{1}, \lambda_{2}, \beta}$ has infinitely many distinct critical point. Thus again, system (1.1) has infinitely many distinct weak solutions. Moreover, Lemma 3.5 implies that the energy of this solutions converges to zero.

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The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that there are no conflicts of interest.