Nonlinear autoregressive sieve bootstrap based on extreme learning machines

1.   Introduction

In the present paper, we proved the existence of a ground state solution of the following nonlinear Choquard equation

on lattice graph $\mathbb {Z}^{N}$. This equation can be viewed as a discrete version of the following Choquard equation

where $\alpha\in(0, N)$, $p > 1$ and $I_{\alpha}:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is the Riesz potential defined at $x\in\mathbb{R}^{N}\setminus\{0\}$ by

with $\Gamma$ being the Euler gamma function.

In the past few decades, many mathematicians have been devoted to studying the Eq (1.2), for example, see ^{[1,2,3,4,5,6]}. In particular, if $N = 3$, $V = 1$ and $p = 2$, i.e., $-\Delta u+u = (I_2*|u|^2)u,$ appeared in the literature at least as early as in 1954's work by Pekar on quantum theory of a Polaron at rest ^[7]. Later in the 1970s, Choquard utilized model (1.2) to describe an electron caught in its own hole, in an approximation to Hartree-Fock theory of one-component plasma ^[1]. Particularly, the equation is also knows as the Schrödinger-Newton equation, which was used to a model of self-gravitating matter ^[8]. Also, the article ^[9] used this system to study the pseudo-relativistic boson stars. In a pioneering work, Lieb ^[1] proved the existence and uniqueness of the ground state to the Eq (1.2) in $\mathbb{R}^{3}$ with $V = 1$, $\alpha = 2$ and $p = 2$. In the paper ^[3], Moroz and Van Schaftingen first obtained the sharp range of the parameter for the existence of solutions of the Eq (1.2) with $\tfrac{N+\alpha}{N} < p < \tfrac{N+\alpha}{N-2}$. If $V$ is the periodic function, since the nonlocal term is invariant under translation, the paper ^[10] got the existence results. Furthermore, Alves ^[11] proved the existence and convergence of nontrivial solutions of the nonlocal Choquard equation. There are tremendous results on this direction in ^{[12,13,14,15,16,17,18]} and the references therein.

On the other hand, the analysis on the graph has become more and more popular, for example, see ^{[19,20,21,22,23,24,25,26,27]}. In a series of work of Grigor'yan et al. ^[19,20,21], they studied the Yamabe type equations, Kazdan-Warner equation and some other nonlinear equations on graph by using the variational methods. In ^[27], Zhang and Zhao investigated the existence of nontrivial solution of the equation $-\Delta u+(\lambda a(x)+1)u = |u|^{p - 1}u$ on the locally finite graphs by using Nehari methods (see ^[28]) and the asymptotic properties of the solution. Later, the paper ^[22] generalized the results of ^[27] to higher order. Furthermore, Hua and Xu ^[] obtained the existence results of nonlinear equation $-\Delta u+V(x)u = f$ on the lattice graph $\mathbb{Z}^{N}$. Recently, Huang et al. investigated extensively the Mean field equation and the relativistic Abelian Chern-Simons equations on the finite graphs by using the variational method in ^[23]. For other related results about the graph, we refer the reader to ^{[29,30,31,32,33,34]} and references therein.

Inspired by the poineering works, in this paper we study the existence and asymptotical behavior of solution for the Choquard equation (1.2) on the lattice graph $\mathbb{Z}^{N}$. For clarity, let us introduce the basic setting on the lattice graph $\mathbb{Z}^{N}$. The graph $\mathbb{Z}^{N}$ consists of the set of vertices

and the set of edges

For any two vertices $x, y \in \mathbb{Z}^{N}$, the distance $d(x, y)$ between them is defined by

where we write $y \sim x$ if and only if the edge $\{x, y\} \in E$. Assume $\Omega \subset\mathbb{Z}^{N}$, we say $\Omega$ is bounded if $d(x, y)$ is uniformly bounded for any $x, y \in \Omega$. It is easy for us to see that a bounded domain of $\mathbb{Z}^{N}$ can contain only finite vertices. We denote the boundary of $\Omega$ is

$C(\mathbb{Z}^{N})$ denotes the set of real-valued functions on $\mathbb{Z}^{N}$. For any $u\in C(\mathbb{Z}^{N})$, its support set is defined as ${\mathop{\rm supp}\nolimits} (u) = \{ {x \in \mathbb{Z}^{N}, u(x)\ne 0} \}$. Let $C_{c}(\mathbb{Z}^{N})$ denote the set of all functions of finite support. We can define the associated gradient for any function $u, v \in C(\mathbb{Z}^{N})$ by

In particular, let $\Gamma(u) = \Gamma(u, u)$ for simplicity. The length of the gradient of $u$ is written by

Let $\mu$ be the counting measure on $\mathbb{Z}^{N}$, i.e., for any subset $A \subset \mathbb{Z}^{N}$, $\mu(A): =$ $\#\{x: x \in A\}$. For any function $f$ on $\mathbb{Z}^{N}$, we write

whenever it makes sense. $\ell^{p}$ is a space endowed with the norm

Assume $u \in C(\mathbb{Z}^{N})$, the Laplacican on $\mathbb{Z}^{N}$ is defined as

The inner product of the Hilbert space $H^{1}(\mathbb{Z}^{N})$ is given by

Therefore, the corresponding norm reads

For a bounded uniformly positive function $V:\mathbb{Z}^{N} \to \mathbb{R}$, it is natural for us to consider the equivalent norm in $H^1(\mathbb{Z}^{N})$ as

Then we have the conclusions for the Eq (1.1).

Theorem 1.1. Let $N \in {\mathbb{N}^*}$, $\alpha\in (0, N)\,$ and $\, p \in (\tfrac{N+\alpha}{N}, \infty)$. Suppose that $V(x):\mathbb{Z}^{N}\to \mathbb{R}$ satisfies the conditions:

(i) $V$ is bounded uniformly positive, i.e. there exist constant ${C_1}, {C_2} > 0$ satisfying ${C_1} < V(x) < {C_2}$ for any $x \in \mathbb{Z}^{N}$.

(ii) $V$ is T-periodic, i.e. for the positive integer $T$, we have $V(x+T e_{i}) = V(x)$, $\forall x \in \mathbb{Z}^{N}, 1 \leq i \leq N$, where $e_{i}$ is the unit vector in the i-th coordinate.

Then there exists a ground state solution of (1.1).

Remark 1.2. The preceding theorem is a discrete version of the results in ^[3]. As in the paper ^[24], we use the Concentration-Compactness Principle (P. L. Lions ^[35,36]) to recover the compactness and prove the existence of ground state solution of (1.1). Interestingly, since the discreteness of the graph, the Sobolev embedding on the lattice graph is different from that in the continuous setting, which allows us to remove the upper critical exponents $\tfrac{N+\alpha}{N-\alpha}$ in the continuous case.

Next we turn to studying the convergence of the solution for the nonlinear Choquard equation. The results of Schrödinger type equation is already considered in the Euclidean space (see ^[11,37]). We may expect that the nonlocal Choquard equation on lattice graphs has some similar results. As the paper ^[22,27], we also consider the confining potential $V = \lambda a(x)+1$, i.e.,

To study the problem (1.3), we introduce the following subspace of ${H^{1}}(\mathbb{Z}^{N})$:

It is easy to recognize that the scalar product of $E_\lambda(\mathbb{Z}^{N})$ is

Then we have the following conclusions.

Theorem 1.3. Let $N \in {\mathbb{N}^*}$, $\alpha\in (0, N)\,$ and $\, p \in [\tfrac{N+\alpha}{N}, \infty)$. Suppose that $a(x):\mathbb{Z}^{N}\to \mathbb{R}$ satisfying

(A1) $a(x)\geq 0$ and the potential well $\Omega = \{x\in \mathbb{Z}^{N}:a(x) = 0\}$ is a non-empty, connected and bounded domain in $\mathbb{Z}^{N}$.

(A2) There exists a point $x_0$ satisfying $a(x) \to \infty$ when $d(x, x_0)\to \infty$.

Then (1.3) has a ground state solution $u_\lambda$ for any constant $\lambda > 1$.

In order to observe the asymptotical properties of $u_\lambda$ as $\lambda \to \infty$, we first study the following Dirichlet problem.

We study the Eq (1.4) in the $H_0^1(\Omega)$ with the norm:

Similarly, the Eq (1.4) also possess a ground state solution.

Theorem 1.4. Let $N \in {\mathbb{N}^*}$, $\alpha\in (0, N)\,$ and $\, p \in (1, \infty)$. Suppose $\Omega$ is a non-empty, connected and bounded domain in $\mathbb{Z}^{N}$. Then the Eq (1.4) has a ground state solution $u\in H_0^1(\Omega)$.

Finally, we show that the solutions $u_\lambda$ of (1.3) converge to a solution of (1.4) as $\lambda\rightarrow \infty$ when the domain in (1.4) is the set of satisfying $a(x) = 0$. On the other words, we obtain the following conclusions.

Theorem 1.5. Let $N \in {\mathbb{N}^*}$, $\alpha\in (0, N)\,$ and $\, p \in [2, \infty)$. Assume that $a(x)$ satisfies $(A1)$ and $(A2)$, then for any sequence $\lambda_k\rightarrow \infty$, up to a subsequence, the corresponding ground state solutions $u_{\lambda_k}$ of (1.3) converge in $H^1(\mathbb{Z}^{N})$ to a ground state solution of (1.4).

The remaining parts of this paper are organized as follows. In Section 2, we give basic definitions and Lemmas on the lattice graph. In Section 3, we establish the discrete Brézis-Lieb Lemma for the nonlocal term and some important conclusions. Section 4 is devoted to proving Theorem 1.1. Then we complete the proof of Theorems 1.3 and 1.4 in Section 5. Finally, we prove Theorem 1.5 in Section 6.

2.   Preliminary results

In this section we give some basic results on the lattice graph. Firstly, we present the formula of integration by parts on lattice graph, which is the basic conclusion when we apply variational methods. Here we omit the concrete proofs and one can refer to ^[22] for more details.

Lemma 2.1. Suppose that $u \in {H^1}({\mathbb{Z}^{N}})$. Then for any $v \in {C_c}({{\mathbb{Z}^{N}}})$, we obtain

Lemma 2.2. Suppose $\Omega \subset \mathbb{Z}^{N}$ is a bounded domain and $u \in {H_0^1}(\Omega)$. Then for any $v \in {C_c}(\Omega)$, we have

Now we are ready to define the weak solution as follows.

Definition 1. Assume $u \in {H^1}({\mathbb{Z}^{N}})$. A function $u$ is called a weak solution of (1.1) if for any $\varphi \in{H^1}({\mathbb{Z}^{N}})$,

Definition 2. Assume $u \in {E_\lambda}({\mathbb{Z}^{N}})$. A function $u$ is called a weak solution of (1.3) if for any $\varphi \in {E_\lambda}(\mathbb{Z}^{N})$,

Definition 3. Assume $u \in H_0^1(\Omega)$. A function $u$ is called a weak solution of (1.4) if for any $\varphi \in H_0^1(\Omega)$,

Notice that if u is a weak solution of (1.1), we infer from Lemma 2.1 that for any test function $\varphi \in H^1(\mathbb{Z}^{N})$,

For any fixed $x_0\in \mathbb{Z}^{N}$, choosing a test function $\varphi : {\mathbb{Z}^{N}}\to\mathbb{R}$ in (2.6) which is defined as

we obtain

which implies that $u$ is a point wise solution of (1.1). Thus, we have the following conclusion for the relationship between the weak solution and the point wise solution.

Proposition 2.3. If $u$ is a weak solution of (1.1), then $u$ is a point wise solution. Similarly, if $u$ is a weak solution of (1.3) or (1.4), then u is also a point wise solution of the corresponding equation.

Finally, we state the following conclusions for the Sobolev embedding.

Lemma 2.4. (^[38]) ${H^1}({\mathbb{Z}^{N}})$ is continuously embedded into ${\ell^q(\mathbb{Z}^{N})}$ for any $q \in [2, \infty]$. Namely, for any $u \in {H^1}(\mathbb{Z}^{N})$, there exists a constant $C_q$ depending only on q such that

Lemma 2.5. ([27,Lemma 2.6]) Assume that $\lambda > 1$ and $a(x)$ satisfies $(A1)$ and $(A2)$. Then $E_\lambda(\mathbb{Z}^{N})$ is continuously embedded into $\ell^q(\mathbb{Z}^{N})$ for any $q\in [2, \infty]$ and the embedding is independent of $\lambda$. Namely, there exists a constant $C_q$ depending only on $q$ such that for any $u \in E_\lambda(\mathbb{Z}^{N})$, $\|u\|_{\ell^q(\mathbb{Z}^{N})} \leq C_q\|u\|_{E_\lambda(\mathbb{Z}^{N})}$. Moreover, for any bounded sequence $\{u_k\} \in E_\lambda(\mathbb{Z}^{N})$, there exists $u \in E_\lambda(\mathbb{Z}^{N})$ such that, up to a subsequence,

Lemma 2.6. ([27,Lemma 2.7]) Assume that $\Omega$ is a bounded domain in $\mathbb{Z}^{N}$. Then $H_0^1(\Omega)$ is continuously embedded into $\ell^q(\Omega)$ for any $q\in [1, \infty]$. Namely, there exists a constant $C_q$ depending only on $q$ such that for any $u \in H_0^1(\Omega)$, $\|u\|_{\ell^q(\Omega)} \leq C_q\|u\|_{H_0^1(\Omega)}$. Moreover, for any bounded sequence $\{u_k\} \in H_0^1(\Omega)$, there exists $u \in H_0^1(\Omega)$ such that, up to a subsequence,

3.   Discrete Brézis-Lieb Lemma

In this section, we give a proof of the discrete Brézis-Lieb Lemma(see ^[3,39,40] for the continuous case) for the nonlocal term on the lattice graph. First, let us recall the discrete Brézis-Lieb Lemma ^[38] for the local case.

Lemma 3.1. ([38,Lemma 9]) Let $\Omega \subset \mathbb{Z}^{N}$ be a domain and $\{u_n\} \subset \ell^{q}(\Omega)$ with $0 < q < \infty$. If $\{u_n\}$ is bounded in $\ell^{q}(\Omega)$ and $u_{n} \rightarrow u$ pointwise on $\Omega$ as $n \to \infty$, then

From Lemma 3.1 and [38,Corollary 10], it is not hard for us to get the following corollary.

Corollary 3.2. Assume $V$ is a uniformly bounded positive function. If $\{u_n\}$ is bounded in $H^1(\mathbb{Z}^{N})$ and $u_{n} \rightarrow u$ pointwise on $\mathbb{Z}^{N}$, then

Next, we prove a variant of the discrete Brézis-Lieb Lemma.

Lemma 3.3. Let $\Omega \subset \mathbb{Z}^{N}$ be a domain, $1 \leq q < \infty$. If the sequence $\{u_n\}$ is bounded in $\ell^{r}(\Omega)$ and $u_{n} \rightarrow u$ pointwise on $\Omega$ as $n \to \infty$, then for every $q\in [1, r]$,

Proof. Applying the Fatou's Lemma, we obtain

Fix $\varepsilon > 0$ and for all $a, b \in \mathbb{R}$, there exists $C_\varepsilon$ satisfying

Hence we obtain

Thus

It follows from the Dominated Convergence Theorem that

From the definition of $f_n^{\varepsilon}$, we obtain

Moreover, one deduces from the basic inequality ${(a+b)^p}\le{C_p}({a^p}+{b^p})(\forall a, b, p > 0)$ that

Therefore, from (3.6) and (3.7), we get

Then let $\varepsilon \rightarrow 0$,

This finishes the proof.

Next, we state the discrete Brézis-Lieb type Lemma.

Lemma 3.4. Suppose $\Omega \subset \mathbb{Z}^{N}$ and $1 \leq p < \infty$. If the sequence $\{u_n\}$ is bounded in $\ell^{p}(\Omega)$ and $u_{n} \rightarrow u$ pointwise on $\Omega$ as $n \to \infty$, then for every $x\in \mathbb{Z}^{N}$, we have

Proof. Since $x\neq y$ and $x, y\in \mathbb{Z}^{N}$, we obtain $|x-y|\geq1$ and it follows that

Thus the proof is complete as $n\to \infty$ from Lemma 3.3.

Now we are in position to establish the discrete Brézis-Lieb Lemma for the nonlocal term of the functional. To this purpose we first present an important inequality on the lattice graph which is studied by many authors in the continuous setting.

Lemma 3.5. (^[41]) (Discrete Hardy-Littlewood-Sobolev Inequality) Let $0 < \alpha < N$, $1 < r, s < \infty$ and $\tfrac{1}{r} + \tfrac{1}{s} + \tfrac{{N - \alpha }}{N} \ge 2$. Assume $f \in \ell^r(\mathbb{Z}^{N})$ and $g \in \ell^s(\mathbb{Z}^{N})$. Then there exists a positive constant $C_{r, s, \alpha}$ depending only on $r, s, \alpha$ such that

The paper ^[38] also give the following equivalent form of (3.9).

Lemma 3.6. Let $0 < \alpha < N$, $1 < r, t < \infty$ and $\tfrac{1}{t} + \tfrac{\alpha}{N}\leq\tfrac{1}{r}$. Assume $f \in \ell^r(\mathbb{Z}^{N})$, then there exists a positive constant $C_{r, t, \alpha}$ depending only on $r, t, \alpha$ such that

The next lemma states the discrete Brézis-Lieb Lemma for the nonlocal term.

Lemma 3.7. Let $1 \leq p < \infty$ and the sequence $\{u_n\}$ is bounded in $\ell^{\frac{2Np}{N+ \alpha}}(\mathbb{Z}^{N})$. Suppose $u_n \rightarrow u$ pointwise on ${\mathbb{Z}^{N}}$ as $n \to \infty$, then

Proof. For every $n$, we can divide the left-hand side of (3.11) into two parts,

where

By Lemma 3.3, taking $q = p$, $r = \tfrac{{2Np}}{{N + \alpha }}$, one has

We first give the estimate for the term ${J_1}$. From the Hardy-Littlewood-Sobolev inequality (Eq 3.9), one deduces that

From (3.13) and $\|u\|_{\ell^\frac{2Np}{N+\alpha}(\mathbb{Z}^{N})}\le \mathop {\liminf }\limits_{n \to \infty }\|u_n\|_{\ell^\frac{2Np}{N+\alpha}(\mathbb{Z}^{N})} < \infty$, it gives that

Now we give the estimate for ${J_2}$. From the Banach-Alaoglu theorem, ${\left|u_n-u\right|^p} \rightharpoonup 0$ weakly in ${\ell^{\frac{2N}{N+\alpha}}}(\mathbb{Z}^{N})$ as ${n \rightarrow \infty }$ and (3.9), we deduce that

We infer from (3.10) that

Moreover, $|u_n-u|^p\rightharpoonup0$ in $\ell^{\frac{2N}{N+\alpha}}(\mathbb{Z}^{N})$. Hence we know that

Then one deduces from (3.13) that $\mathop {\lim }\limits_{n \to \infty } {J_2} = 0$. This together with (3.14), we get the results.

4.   Proof of the Theorem 1.1

In the present section we are devoted to the proof of Theorem 1.1. Obviously, for any function $u:\mathbb{Z}^{N} \to \mathbb{R}$, the energy functional related to (1.1) is given by

Notice that the functional $J$ is well defined in $H^1(\mathbb{Z}^{N})$. Indeed, assume that $u \in \ell^{\frac{2NP}{N+\alpha}}(\mathbb{Z}^{N})$, then by applying the Hardy-littlewood-Sobolev inequality (Eq 3.9) to the function $f = |u|^p\in {\ell^{\frac{2N}{N + \alpha}}}(\mathbb{Z}^{N})$, we obtain

It sufficient for us to confirm when the condition $u \in \ell^{\frac{2NP}{N+\alpha }}(\mathbb{Z}^{N})$ is satisfied. According to the Lemma 2.4, $H^1(\mathbb{Z}^{N})$ is continuously embedded into $\ell^{\frac{2Np}{N+\alpha }}(\mathbb{Z}^{N})$ if and only if $p \ge \tfrac{N+\alpha}{N}$. Moreover, we infer from the inequality (Eq 3.9) that

where the constant $C_{N, p, \alpha}$ depends only on $N, \alpha$ and $p$. Based on the previous argument, the function $J$ is meaningful.

Next, we define the Nehari manifold related to (4.1) by

Let

If there exists a function $u \in \mathcal{N}$ satisfying $J(u) = m$, then the function $u$ is called a ground state solution. Obviously, $u$ is a critical point of $J$.

Next, we shall find the critical point of the functional (4.1).

Proposition 4.1. Let $N \in\mathbb{N}^*$, $\alpha \in (0, N)$ and $p \in (1, \infty)$. If $u \in {H^1}(\mathbb{Z}^{N}) \cap {\ell^{\frac{2Np}{N+\alpha}}}(\mathbb{Z}^{N}) \setminus \{0 \}$ and $V$ is a uniformly bounded positive function, there holds

where

Proof. For any $t > 0$, we set

By a direct computation,

When $s'(t) = 0$, we can obtain a unique ${t_u}$ such that $s'(t_u) = 0$. Moreover, one has

Since as $0 < t < {t_u}$, $s'(t) > 0$ and as $t > {t_u}$, $s'(t) < 0$, thus

This finishes the proof.

Note that the ground state energy of $J$ can be characterized as

In the next conclusion we show the infirmum of $S(u)$ can be achieved by some nontrivial function.

Proposition 4.2. Let $N \in\mathbb{N}^*$, $\alpha \in (0, N)$ and $p \in \left(\tfrac{N+\alpha}{N}, \infty \right)$. Suppose that $V$ is a uniformly bounded positive function, then there exists $u \in {H^1}(\mathbb{Z}^{N})$ satisfying

Combining with Propositions 4.1 and 4.2, we complete the proof of Theorem 1.1. Then we only need to focus on the proof Proposition 4.2 in the next. In the Euclidean space, we are familiar with the different kinds of the proof of Proposition 4.2. For example, a strategy consists in minimizing among radial functions and then prove with the symmetrization by rearrangement that a radial minimizer is a global minimizer. In our setting, the main difficulty for the analysis is that there is no proper counterpart for radial functions on $\mathbb{Z}^{N}$ and moreover we do not have the compactness in this problem. To overcome the difficulty we borrow an idea of [42,Section 4](also see ^[24]) and use the constraint method to prove Proposition 4.2.

Proof of Proposition $4.2$. Set

then we can get

Let $\{u_n\}$ be a minimizing sequence in ${H^1}(\mathbb{Z}^{N})$ such that

and $\mathop{\lim}\limits_{n\rightarrow \infty}\frac{1}{S(u_n)} = \frac{1}{m}$. By the discrete Hardy-Littlewood-Sobolev inequality (Eq 3.9), we obtain

Taking the limit from both sides, one can see

Since $p > \tfrac{N+\alpha}{N}$, we obtain

Hence, there exists a subsequence $\{u_n\}$ and a sequence $\{y_n\} \subset \mathbb{Z}^{N}$ such that $| {u_n(y_n)}| \ge C$ for each $n$. By translations, we define $\widetilde{u}_n = :{u_n}(y+{k_n}T)$ with ${k_n} = (k_n^1, \cdots k_n^N)$ to ensure that $(y_n-{k_n}T) \subset \Omega$ where $\Omega = [0, T)^N \cap \mathbb{Z}^{N}$ is a bounded domain in $\mathbb{Z}^{N}$. Then for each $\widetilde{u}_n$,

Moreover, by translation invariance, we infer from $V(x)$ is T-periodic in $x$ that

and

Without loss of generality, we can get a minimizing sequence $\{u_n\}$ satisfying ${\|u_n\|_{{\ell^\infty }(\Omega)}} \ge C > 0$. Since $\Omega$ is bounded, there exists at least one point, say $x_0$, such that ${u_n}(x_0)\to u(x_0)\ge C > 0$. Since the sequence $\{u_n\}$ is bounded in ${H^1}(\mathbb{Z}^{N})$, it follows that ${u_n} \rightharpoonup u$ in ${H^1}(\mathbb{Z}^{N})$ and ${u_n} \to u\ne 0$ pointwise on $\mathbb{Z}^{N}$. Then it follows from Corollary 3.2 and Lemma 3.7 that

For every $n$, we have

Since $u\neq0$, one has

which yields

By (4.7), one has

By Fatou's Lemma, one gets

Then it is enough for us to prove that $\int_{\mathbb{Z}^{N}} {\left({|\nabla u_n|^2}+V(x)u_n^2 \right)} d\mu = 1$. Using a contradiction argument, suppose that

then by

However, $(a+b)^p > a^p+b^p$ if $a, b > 0$. This yields a contradiction by (4.7).

5.   Proof of Theorems 1.3 and 1.4

In this section we shall prove the existence result for (1.3) and (1.4) by using the standard variational methods. Obviously, the functional associated with the problem (1.3) is given by

where $p \ge \tfrac{N+\alpha}{N}$. The corresponding Nehari manifold is defined as

We define the least energy level $m_\lambda$ by

Then we first prove the Nehari manifold $\mathcal{N_\lambda}$ is nonempty.

Lemma 5.1. The Nehari manifold $\mathcal{N_\lambda}$ is non-empty.

Proof. For $t\in\mathbb{R}$ and fix a function $u \in E_\lambda(\mathbb{Z}^{N})\backslash\{0\}$ and, we define

Since $p > 1$ and $u\neq 0$, it is obvious that $\gamma(t) > 0$ for small $t > 0$ and that $\lim\limits_{t \rightarrow \infty}\gamma(t) = -\infty$. Then there exists $t_0 \in (0, \infty)$ such that $\gamma(t_0) = 0$, which implies that $t_0u \in\mathcal{N_\lambda}$.

Next, we prove the least energy level $m_\lambda$ is positive.

Lemma 5.2. We have $m_\lambda = \inf\limits_{u \in \mathcal{N_\lambda}} J_\lambda(u) > 0$.

Proof. Since $u \in \mathcal{N_\lambda}$, then

By Lemma 2.5 and (3.9), we obtain

where $C$ is independent of $\lambda$. It follows from $p > 1$ that

This gives

The next lemma states that the least energy $m_\lambda$ can be achieved.

Lemma 5.3. The value $m_\lambda$ can be achieved by some $u_\lambda \in \mathcal{N_\lambda}$. Namely, there exists some $u_\lambda \in \mathcal{N_\lambda}$ such that $J_\lambda(u_\lambda) = m_\lambda$.

Proof. Take a minimizing sequence $\{u_k\} \subset \mathcal{N_\lambda}$ such that $\mathop{\lim}\limits_{k\rightarrow \infty}{J_\lambda(u_k)} = m_\lambda$. Since

we have that $\{u_k\}$ is bounded in $E_\lambda(\mathbb{Z}^{N})$, where $\mathop{\lim}\limits_{k\rightarrow \infty}{o_k(1)} = 0$. By Lemma 2.5, we can assume that there exists some $u_\lambda \in E_\lambda(\mathbb{Z}^{N})$ such that

From the discrete Hardy-Littlewood-Sobolev inequality (Eq 3.9), we infer that

Therefore, one has

Then from the Lemma 3.7, we get

Since the $E_\lambda$ norm is weakly lower semi-continuous, one has

Next it suffices to show that $u_\lambda\in \mathcal{N_\lambda}$. We infer from (5.1) that

This together with (5.2) which implies that

Therefore $u_\lambda\neq0$. Since $u_k \in \mathcal{N_\lambda}$, we infer that

We use the contradiction argument to obtain our results. Assume that

Similar as the proof of Lemma 5.1, there would exist a $t\in(0, 1)$ such that $tu_\lambda \in \mathcal{N_\lambda}$. This implies that

This contradicts the fact that $m_\lambda = \mathop{\inf}\limits_{u \in \mathcal{N_\lambda}} J_\lambda(u)$. Therefore we have $u_\lambda \in \mathcal{N_\lambda}$. Moreover, we infer from (5.3) that $m_\lambda$ is achieved by $u_\lambda$.

The following Lemma finishes the proof of Theorem 1.3.

Lemma 5.4. $u_\lambda \in \mathcal{N_\lambda}$ is a critical point for $J_\lambda$.

Proof. It is enough for us to prove that for any $\phi \in E_\lambda(\mathbb{Z}^{N})$, there holds

Since $u_\lambda \not\equiv 0$, we can choose a constant $\varepsilon > 0$ such that $u_\lambda+s\phi \not\equiv 0$ for all $s \in (-\varepsilon, \varepsilon)$. Furthermore, for every given $s \in (- \varepsilon, \varepsilon)$, we can find some $t(s) \in (0, \infty)$ satisfying $t(s)(u_\lambda+s\phi) \in \mathcal{N_\lambda}$. Indeed, $t(s)$ can be taken as

Obviously, we can get $t(0) = 1$. Take a function $\gamma(s) :(-\varepsilon, \varepsilon) \rightarrow \mathbb{R}$ which is defined as

For $t(s)(u_\lambda+s\phi) \in \mathcal{N_\lambda}$ and $J_\lambda(u_\lambda) = \mathop{\inf}\limits_{u\in\mathcal{N_\lambda}} J_\lambda(u)$, $\gamma(s)$ achieves its minimum at $s = 0$. Together with $u_\lambda\in\mathcal{N_\lambda}$ and $J'_\lambda (u_\lambda)u_\lambda = 0$, it follows that

Next we focus on the proof of Theorem 1.4. The functional associated with the Eq (1.4) is given by

We remark that $\|u\|_{\ell^q(\Omega)} \leq C\|u\|_{H_0^1(\Omega)}$ for $q\in [1, \infty]$ by Lemma 2.6. Therefore, the functional $J_\Omega(u)$ is well defined as $p\geq\frac{N+\alpha}{2N}$. The corresponding Nehari manifold is defined as

Let $m_\Omega$ be

Since $\Omega$ contains only finite vertices, the proofs of the previous results can be easily applied to the Eq (1.4). Moreover, $p > 1$ is enough for us to prove Theorem 1.4. Here we omit the details of the proofs.

6.   Convergence of the ground state solution

In the current section, we mainly focus on the asymptotical properties of the solution. That is, we show that the ground state solutions $u_\lambda$ of (1.3) converge to a ground state solution of (1.4) as $\lambda\rightarrow \infty$. To accomplish this we first prove that any solution of (1.3) is bounded away from zero.

Lemma 6.1. There exists a constant $\sigma > 0$ which is independent of $\lambda$, such that for any critical point $u\in E_\lambda(\mathbb{Z}^{N})$ of $J_\lambda$, we have $\|u\|_{E_\lambda(\mathbb{Z}^{N})}\geq \sigma$.

Proof. From Lemma 2.5 and the inequality (Eq 3.9), one has

where $C$ is independent of $\lambda$. Then we can choose $\sigma = (\frac{1}{C})^{\frac{p}{p-1}}$ and Lemma 6.1 is proved.

The next lemma studies the property of $(PS)_c$ sequence of $J_\lambda$.

Lemma 6.2. For any $(PS)_c$ sequence $\{u_k\}$ of $J_\lambda$, there holds

Furthermore, there would exist a constant $C_1 > 0$ independent of $\lambda$, such that either $c\geq C_1$ or $c = 0$.

Proof. Since $J_\lambda(u_k) \to c$ and $J'_\lambda(u_k) \to 0$ as $k \to \infty$, we have

which gives (6.1). By Lemma 2.5 and (3.9), for any $u\in E_\lambda(\mathbb{Z}^{N})$, we obtain

Take $\rho = (\frac{1}{2C^{2p}})^\frac{1}{2p-2}$. If $\|u\|_{E_\lambda(\mathbb{Z}^{N})}\leq \rho$, we get

Take $C_1 = \frac{p-1}{2p}\rho^2$ and suppose $c < C_1$. Since $\{u_k\}$ is a $(PS)_c$ sequence, it yields

Hence, for large $k$, we have

which implies that $\|u_k\|_{E_\lambda(\mathbb{Z}^{N})}\to 0$ as $k \to \infty$. It follows immediately that $J_\lambda(u_k) \to c = 0$ and the positive constant can be taken as $C_1 = \frac{p-1}{2p}\rho^2 = (\frac{1}{2C^{2p}})^\frac{1}{p-1}$.

Remark 6.3. If we take $c = m_\lambda$, then there would exist a $(PS)_c$ sequence ${u_k}$ such that $u_k\rightharpoonup u_\lambda$ when proving the existence of a ground state solutions $u_\lambda$. Since the $E_{\lambda_k}$ norm of ${u_{\lambda_k}}$ is weakly lower semi-continuous, then $\|u_\lambda\|_{E_\lambda(\mathbb{Z}^{N})}$ is bounded by $\tfrac{2p}{p-1} m_\lambda$.

Next, we study the relationship between the ground states $m_\lambda$ and $m_\Omega$.

Lemma 6.4. $m_\lambda \to m_\Omega$ as $\lambda \to \infty$.

Proof. Notice that $m_\lambda \leq m_\Omega$ for every positive $\lambda$ owing to $\mathcal{N}_\Omega \subset \mathcal{N}_\lambda$. Take a sequence $\lambda_k \to \infty$ satisfying

where $m_{\lambda_k}$ is the ground state and $u_{\lambda_k}\in \mathcal{N}_{\lambda_k}$ is the corresponding ground state solution of (1.3). Then it follows $M > 0$ from Lemma 6.2. According to Remark 6.3, we know that the $E_{\lambda_k}$ norm of ${u_{\lambda_k}}$ is controlled by the constant $\tfrac{2p}{p-1} m_\Omega$, which is independent of $\lambda_k$. Up to a subsequence, we can assume that $u_{\lambda_k}(x)\rightarrow u_0(x)$ on $\mathbb{Z}^{N}$ and for any $q \in [2, +\infty)$, $u_{\lambda_k}\rightarrow u_0$ in $\ell^q(\mathbb{Z}^{N})$. Moreover, we get that $u_0\not\equiv 0$ from Lemma 6.1.

We first claim that $u_0|_{\Omega^c} = 0$. If it is not true, we can find a point $x_0$ satisfying $u_0(x_0) \neq0$. Since $u_{\lambda_k}\in \mathcal{N}_{\lambda_k}$, then

Since $a(x_0) > 0, u_{\lambda_k}(x_0) \to u_0(x_0)\neq0$ and $\lambda_k \to \infty$, we get

which contradicts with the conclusion $m_{\lambda_k} \leq m_\Omega$. Since the norm $\|\cdot\|_{H^1(\mathbb{Z}^{N})}$ is weakly lower semi-continuous and (5.2), we get

Noticing that $u_0|_{\Omega^c} = 0$, we get

Then there exists $\alpha\in(0, 1]$ such that $\alpha u_0 \in \mathcal{N}_\Omega$, i.e.,

This implies that

Consequently, $M \geq m_{\Omega}$. Combining with (6.3), we get that

Next, we are devoted to proving Theorem 1.5.

Proof of Theorem 1.5. We need to prove that for any sequence $\lambda_k \to \infty$, the corresponding $u_{\lambda_k} \in\mathcal{N}_{\lambda_k}$ satisfying $J_{\lambda_k}(u_{\lambda_k}) = m_{\lambda_k}$ converges in $H^1(\mathbb{Z}^{N})$ to a ground state solution $u_\Omega$ of (1.4) along a subsequence. According to Remark 6.3, the $E_{\lambda_k}$ norm of ${u_{\lambda_k}}$ is uniformly bounded by the constant $\tfrac{2p}{p-1} m_\Omega$, which is independent of $\lambda_k$. Consequently, we can assume that there would exist some $u_0$ satisfying $u_{\lambda_k}(x)\rightarrow u_0(x)$ in $\mathbb{Z}^{N}$and for any $q \in [2, +\infty)$, $u_{\lambda_k}\rightarrow u_0$ in $\ell^q(\mathbb{Z}^{N})$. Moreover, we get that $u_0\not\equiv 0$ from Lemma 6.1. As what we have done in Lemma 6.4, we can prove that $u_0|_{\Omega^c} = 0$.

First, we claim that

and

If for some $\delta > 0$, there holds

we have

Then there exists $\alpha\in(0, 1)$ such that $\alpha u_0 \in \mathcal{N}_\Omega$. On the other hand, if

we also have $\int_{\Omega \cup \partial\Omega}\left({|\nabla u_0|}^2+{u_0^2}\right)d\mu < \int _{\Omega}\left(\sum\limits_{y\ne x \atop y \in \Omega } {\frac{|u_0(y)|^p}{|x - y|^{N-\alpha}}}\right){|u_0|}^p d\mu$. Then in both cases, we can find $\alpha\in(0, 1)$ such that $\alpha u_0 \in \mathcal{N}_\Omega$. Consequently, we have

which arrives at a contradiction.

To prove Theorem 1.5, we also need verify that $u_0$ is a ground state solution of (1.4). The first step is to prove that $u_0$ is a critical point of $J_\Omega$. Since $J'_{\lambda_k}(u_{\lambda_k})\phi = 0$, for any $\phi \in H_0^1(\Omega)\subset H^1(\mathbb{Z}^{N})$, we have

Since $a(x) = 0$ in $\Omega$ and $\phi = 0$ in $\Omega^c$, there holds

Let $k\to \infty$, the above equality becomes

Since $u_{\lambda_k}\rightarrow u_0$ in $\ell^p(\mathbb{Z}^{N})$ with $p\geq2$ and Lemma 3.4, we obtain

which yields $u_0\in\mathcal{N}_\Omega$, and $u_0$ is a solution of (1.4).

Finally, we prove that $u_0$ achieves the infimum of $J_\Omega$ in $\mathcal{N}_\Omega$.

Since $J_{\lambda_k}(u_{\lambda_k}) = m_{\lambda_k}$, we get $J_{\Omega}(u_0) = m_{\Omega}$ by Lemma 6.4. Hence the function $u_0$ is a ground state solution of (1.4).

Finally, we have the following lemma for the convergence of the sequence $\{u_{\lambda_k}\}$.

Corollary 6.5. Furthermore, we have $\lim\limits_{k \to \infty}\|u_{\lambda_k}-u_0\|_{E_{\lambda_k}(\mathbb{Z}^{N})} = 0$.

Proof. Indeed, since $u_{\lambda_k} \in\mathcal{N}_{\lambda_k}$ and $u_0|_{\Omega^c} = 0$, we have

which finishes the proof.

Acknowledgments

This work was supported by NNSF of China (Grants 11971202), Outstanding Young foundation of Jiangsu Province No. BK20200042.

Conflict of interest

The authors declare no conflict of interest.