- Research
- Open access
- Published:
Ground-state sign-changing homoclinic solutions for a discrete nonlinear p-Laplacian equation with logarithmic nonlinearity
Boundary Value Problems volume 2024, Article number: 6 (2024)
Abstract
By using a direct non-Nehari manifold method from (Tang and Cheng in J. Differ. Equ. 261:2384–2402, 2016), we obtain an existence result of ground-state sign-changing homoclinic solutions that only changes sign once and ground-state homoclinic solutions for a class of discrete nonlinear p-Laplacian equations with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground-state energy is larger than twice the ground-state energy.
1 Introduction
The existence of solutions for the discrete nonlinear p-Laplacian equations by variational methods has been a hot topic in the last twenty years and we refer readers to [4, 13, 15, 19, 20] for example. In particular, in [4], Chen-Tang considered the following discrete p-Laplacian system:
where \(p>1\), \(\varphi _{p}\) is the p-Laplace operator, \(u\in \mathbb{R}^{N}\), \(a:\mathbb{Z}\to \mathbb{R}\) and \(W:\mathbb{Z}\times \mathbb{R}^{N}\to \mathbb{R}\). When \(W(n,x)\) is an odd function in x, continuously differentiable, and satisfies other suitable conditions, they obtained that the system has an unbounded sequence of homoclinic solutions using the symmetric mountain-pass theorem. When \(p=2\), (1.1) reduces to the discrete nonlinear Schrödinger (DNLS) equation. The DNLS equation is one of the most important inherently discrete models and plays a crucial role in modeling various phenomena from solid-state and condensed-matter physics to biology [7–10]. In recent years, the existence of standing-wave solutions for the DNLS equation has attracted some attention (see [3, 5, 11, 14, 23]). In particular, in [5], Chen-Tang-Yu studied the following DNLS equation:
When f satisfies the superquadratic growth condition and the monotonicity condition, using the method in [6] and [21], they obtained that the equation has a ground-state solution and a least-energy sign-changing solution, which changes sign exactly once. Furthermore, they obtained that the energy of the sign-changing solution is twice that of the ground-state solution. Next, we recall two studies [2] and [21] that inspire our work partially. In [2], Chang-Wang-Yan studied the following logarithmic Schrödinger equation on a locally finite graph \(G=(V, E)\):
where \(a:V\to \mathbb{R}\). When a is bounded from below and the volume of set \(\{x\in V: a(x)\leq M\}\) is finite, they used the Nehari manifold method to obtain that the equation has a ground-state solution. Moreover, when a is bounded from below and \(1/a(x)\) is a Lebesgue integrable function on the set \(\{x\in V: a(x)> M_{0}\}\), they also found that the equation has a ground-state solution by using the mountain-pass theorem. In [21], Tang-Cheng investigated the following Kirchhof-type problem:
where Ω is a bounded domain in \(\mathbb{R}^{N}\), \(N=1,2,3\). When f satisfies the supercubic growth and the monotonicity condition, they used a new energy inequality, the deformation lemma, Miranda’s theorem, and the non-Nehari manifold method to obtain the same result as in [5].
In this paper, inspired by [2, 4, 5], we mainly use the method in [21] to develop the results in [5] to the following discrete nonlinear p-Laplacian equation involving logarithmic nonlinearity:
where \(1< p< q\), \(\varphi _{p}(s)=|s|^{p-2}s\) is the p-Laplacian operator, \(\frac{p}{2}\in \mathbb{N}^{*}\), \(\mathbb{N}^{*}\) denotes the positive integer set, \(a, b, c:\mathbb{Z}\rightarrow (0,+\infty )\), \(r\geq 1\), \(u:\mathbb{Z}\to \mathbb{R}\), and \(\Delta u(n)=u(n+1)-u(n)\) is the forward difference operator. Note that the nonlinear term \(c(n)|u(n)|^{q-2}u(n)\ln{|u(n)|^{r}}\) does not satisfy the monotonicity condition in [5]. Therefore, the situation we studied is different from that in [5] even if \(p=2\). There exist two main difficulties in studying equation (1.2). One is that the associated functional I of equation (1.2) is not well defined in E, which is caused by the logarithmic nonlinearity, and the other is that the quasilinearity of the p-Laplacian operator makes it difficult and complex to establish energy inequalities. For the first difficulty, we mainly use the idea in [2] to establish a well-defined space \(\mathcal{D}\), thereby avoiding the case that \(\sum_{n\in \mathbb{Z}}c(n)|u(n)|^{q}\ln{|u(n)|^{r}}=-\infty \). For the second difficulty, we use the binomial theorem and the combination number formula, and then by some careful calculations and analysis, establish some useful energy inequalities. We introduce the following assumptions:
- \((C_{1})\):
-
there exists a positive constant \(b_{0}\) such that \(b(n)\geq b_{0}\) for all \(n\in \mathbb{Z}\) and \(\lim_{|n|\rightarrow +\infty} b(n)=+\infty \);
- \((C_{2})\):
-
there is a positive constant \(c_{0}\) such that \(c(n)\leq c_{0}\) for all \(n\in \mathbb{Z}\) and \(\sum_{n\in \mathbb{Z}}c(n)<+\infty \).
Next, we define
and
Then, E is a reflexive Banach space. As usual, let \(1< p<+\infty \) and define
with the norm
When \(p=+\infty \), we define
with
Note that equation (1.2) is formally related to the energy functional \(I: E\rightarrow \mathbb{R}\cup \{+\infty \}\) that is defined by
However, the functional I is not well defined in E (see Appendix 1). We discuss the functional I on the set
that is,
Note that
for all \(n\in \mathbb{Z}\), where \(\zeta \in (q,+\infty )\). Then, by \((C_{2})\), for any given \(\varepsilon >0\), there exists a positive constant \(C_{\varepsilon}\) such that
Then, \(\mathcal{D}\) is the closed subspace of E, \(I\in C^{1}(\mathcal{D},\mathbb{R})\) and
Using Abel’s partial summation formula (also known as Abel’s transformation) in [16] and the definition of \(\Delta u(n)\), we have
which implies that
According to the above equations, we can derive that \(\langle I'(u),v\rangle =0\) for any \(v\in \mathcal{D}\) if and only if
Therefore, it is easy to see that the critical points of I in \(\mathcal{D}\) are solutions of equation (1.2). Furthermore, if \(u\in \mathcal{D}\) is a solution of equation (1.2) and \(u^{\pm}\neq 0\), then u is a sign-changing solution of equation (1.2), where
To be precise, we obtain the following results.
Theorem 1.1
Assume that \((C_{1})\) and \((C_{2})\) hold. Then, problem (1.2) has a sign-changing solution \(u_{0}\in \mathcal{M}\) such that \(I(u_{0})=\inf_{\mathcal{M}}I:=m_{*}>0\) and \(u_{0}\) only changes the sign once, where
Theorem 1.2
Assume that \((C_{1})\) and \((C_{2})\) hold. Then, problem (1.2) has a solution \(\bar{u}\in \mathcal{N}\) such that \(I(\bar{u})=\inf_{\mathcal{N}}I:=c_{*}>0\), where
In addition, \(m_{*}\geq 2c_{*}\).
2 Preliminaries
In this section, we provide some lemmas that play some important roles in the proofs of our results.
Lemma 2.1
Assume that \((C_{1})\) holds. Then, \(\mathcal{D}\) is continuously embedded into \(l^{\kappa}(\mathbb{Z},\mathbb{R})\) for any \(p\leq \kappa \leq +\infty \), that is, for all \(u\in \mathcal{D}\),
Moreover, \(\mathcal{D}\) is compactly embedded in \(l^{\kappa}(\mathbb{Z},\mathbb{R})\) for any \(p\leq \kappa \leq +\infty \).
Proof
For any \(u\in \mathcal{D}\), when \(\kappa =p\), there holds
When \(\kappa =+\infty \), we can also obtain that
For any \(p<\kappa <+\infty \), it follows from (2.2) that
Hence, (2.1) holds.
Next, we prove that the embeddings are also compact. Suppose that \(\{u_{k}\}\) is a bounded sequence in \(\mathcal{D}\). Then, there is a subsequence of \(\{u_{k}\}\), still denoted by \(\{u_{k}\}\), such that \(u_{k}\rightharpoonup u\) weakly in \(\mathcal{D}\) for some point \(u\in \mathcal{D}\). In particular,
where \(\varphi \in \mathcal{D}\) is defined by
for any fixed n. Thus, we have
We now prove \(u_{k}\rightarrow u\) in \(l^{\kappa}(\mathbb{Z},\mathbb{R})\) for all \(p\leq \kappa \leq +\infty \). When \(\kappa =p\), since \(u\in \mathcal{D}\), according to the boundedness of \(\{u_{k}\}\) and the definition of \(\|\cdot \|\), there appears a positive constant \(\delta _{0}\) such that
For any given positive constant \(\varepsilon _{1}\), there is a \(n_{0}\in \mathbb{Z}\) such that \(\frac{1}{b(n)}<\varepsilon _{1}\) as \(|n|>n_{0}\). Therefore, we can obtain that
On the other hand, (2.3) implies that \(\lim_{k\rightarrow +\infty}\sum_{|n|\le n_{0}}|u_{k}(n)-u(n)|^{p}=0\) since \(\{n\in \mathbb{Z}:|n|\le n_{0}\}\) is a finite set. Then, according to the arbitrariness of \(\varepsilon _{1}\) and (2.4), we have
For \(\kappa =+\infty \), according to the definition of \(\|\cdot \|_{l^{\infty}}\) and (2.5), as \(k\rightarrow +\infty \), we have
and for \(p<\kappa <+\infty \), by (2.5) and (2.6), there exists
Consequently, by (2.5), (2.6), and (2.7), we can derive that \(u_{k}\rightarrow u\) in \(l^{\kappa}(\mathbb{Z},\mathbb{R})\) for all \(p\leq \kappa \leq +\infty \). □
Proposition 2.1
For all \(\frac{p}{2}\in \mathbb{N}^{*}\) and \(u\in \mathcal{D}\), there hold
and
Proof
Let
Note that \(\Delta u^{+}(n)\Delta u^{-}(n)=-u^{+}(n+1)u^{-}(n)-u^{+}(n)u^{-}(n+1) \geq 0\). Then, according to the definition of \(\|\cdot \|\), Appendix 1 below, and the binomial theorem, we have
Similarly, we have
and
By (1.4), (1.7), (2.8), (2.9), and (2.10), it is easy to see that the conclusions hold. □
Next, we establish an inequality associated to \(I(u)\), \(I(su^{+}+tu^{-})\), \(\langle I'(u),u^{+}\rangle \), and \(\langle I'(u),u^{-}\rangle \).
Lemma 2.2
Assume that \((C_{1})\) and \((C_{2})\) hold. For all \(u\in \mathcal{D}\) and \(s,t\ge 0\), there exists
where \(\Theta = \frac{2s^{p}C_{\frac{p}{2}-1}^{i} C_{i}^{j}+s^{p}C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j}+2t^{p}C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j-1} +t^{p}C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j}-2s^{p-(i+j)}t^{i+j}C_{\frac{p}{2}}^{i} C_{i}^{j}}{2p} \geq 0\).
Proof
It is easy to see that (2.11) holds for \(u=0\). Next, we let \(u\neq 0\). According to Appendix 1 in [17], there holds
For \(u\in \mathcal{D}\backslash \{0\}\) and all \(s,t\ge 0\), we have
By virtue of Appendix 2 in [17], the function \(f(x)=\frac{1-a^{x}}{x}\) is strictly monotonically decreasing on \((0,+\infty )\) for \(a>0\) and \(a\neq 1\). Then, by (1.4), (2.8), (2.13), (2.9), (2.10), (1.7), (2.12), and Appendix 2 below, we can derive the following inequality
where \(\Theta '= \frac{2C_{\frac{p}{2}}^{i} C_{i}^{j}-2C_{\frac{p}{2}-1}^{i} C_{i}^{j}-C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j} -2C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j-1}-C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j}}{2p}=0\) (use the combination number formula) and \(\Theta \ge 0\) (see Appendix 2). Hence, we obtain that (2.11) holds for all \(u\in \mathcal{D}\), \(s,t\geq 0\). □
Remark 2.1
Let \(s=t\) in (2.11). It is easy to see that \(\Theta =0\). Then, for all \(u\in \mathcal{D}\) and \(t\geq 0\), there holds
Corollary 2.3
Assume that \((C_{1})\) and \((C_{2})\) hold. For all \(u\in \mathcal{D}\) and \(t\geq 0\), we have
Proof
According to (1.4), (1.7), and (2.12), there exists
Hence, (2.15) holds for all \(u\in \mathcal{D}\) and \(t\geq 0\). □
Note that \(1< p< q\), \(\Theta \geq 0\) and the function \(f(x)=\frac{1-a^{x}}{x}\) is strictly monotonically decreasing on \((0,+\infty )\) for \(a>0\) and \(a\neq 1\). Then, in combination with Lemma 2.2, we have the following corollary.
Corollary 2.4
Assume that \((C_{1})\) and \((C_{2})\) hold. For any \(u\in \mathcal{M}\), we can obtain that \(I(u)=\max_{s,t\geq 0}I(su^{+}+tu^{-})\).
In combination with Corollary 2.3 or Remark 2.1, we have the following corollary.
Corollary 2.5
Assume that \((C_{1})\) and \((C_{2})\) hold. For any \(u\in \mathcal{N}\), there holds \(I(u)=\max_{t\geq 0}I(tu)\).
Lemma 2.6
Assume that \((C_{1})\) and \((C_{2})\) hold. For any \(u\in \mathcal{D}\) with \(u\neq 0\), there exists a unique positive constant \(t_{0}\) such that \(t_{0}u\in \mathcal{N}\).
Proof
First, we prove the existence of \(t_{0}\). For any \(u\in \mathcal{D}\) with \(u\neq 0\), let \(u\in \mathcal{N}\) be fixed and define a function \(g(t)=\langle I'(tu),tu \rangle \) on \((0,+\infty )\). On the one hand, by (1.6) and Lemma 2.1, there exist two positive constants \(\varepsilon _{2}<\frac{b_{0}}{c_{0}}\) and \(C_{\varepsilon _{2}}\) such that
Then, according to \(\zeta >q\) and \(q>p>1\), we have that \(g(t)>0\) for all sufficiently small \(t>0\).
On the other hand, noting that \(c(n)>0\) for all \(n\in \mathbb{Z}\), by \((C_{2})\) and (1.6), there exists
Then, by \(1< p< q\), \(r\ge 1\) and (2.18), it is easy to see \(g(t)<0\) for all large t. Hence, it follows from the continuity of \(g(t)\) that there exists a \(t_{0}\in (0,+\infty )\) such that \(g(t_{0})=0\), which implies that there exists a positive constant \(t_{0}\) such that \(t_{0}u\in \mathcal{N}\).
Next, we prove the uniqueness of \(t_{0}\). Proofing by contradiction, we assume that there exist \(u\in \mathcal{D}\) and two positive numbers \(t_{1}\neq t_{2}\) such that \(t_{1}u\in \mathcal{N}\) and \(t_{2}u\in \mathcal{N}\). Note that the function \(f(x)=\frac{1-a^{x}}{x}\) is strictly monotonically decreasing on \((0,+\infty )\) for \(a>0\) and \(a\neq 1\). Taking u as \(t_{1}u\) and t as \(\frac{t_{2}}{t_{1}}\) in Corollary 2.3, there holds
On the other hand, taking u as \(t_{2}u\) and t as \(\frac{t_{1}}{t_{2}}\) in Corollary 2.3, there also holds
Hence, (2.19) contradicts (2.20). Hence, \(t_{1}=t_{2}\), that is, there exists a unique positive constant \(t_{0}\) such that \(t_{0}u\in \mathcal{N}\). □
Lemma 2.7
Assume that \((C_{1})\) and \((C_{2})\) hold. For any \(u\in \mathcal{D}\) with \(u^{\pm}\neq 0\), there exists a unique pair of positive constants \((s_{0},t_{0})\) such that \(s_{0}u^{+}+t_{0}u^{-}\in \mathcal{M}\).
Proof
First, we prove the existence of \((s_{0},t_{0})\). For any \(u\in \mathcal{D}\) with \(u^{\pm}\neq 0\), according to (2.9) and (2.10), we have
and
It follows from (2.17) and (2.18) that \(h_{1}(s,s)>0\) and \(h_{2}(s,s)>0\) for \(s>0\) sufficiently small and \(h_{1}(t,t)<0\) and \(h_{2}(t,t)<0\) for \(t>0\) large enough. Thus, there are two constants \(0<\theta _{1}<\theta _{2}\) such that
For all \(s,t\in [\theta _{1},\theta _{2}]\), according to (2.21), (2.22), and (2.23), there exists
and similarly, we can obtain that
Therefore, by virtue of (2.24), (2.25), and the Pincaré–Miranda Theorem [12], there appears a point \((s_{0},t_{0})\) with \(\theta _{1}< s_{0},t_{0}<\theta _{2}\) such that \(h_{1}(s_{0},t_{0})=h_{2}(s_{0},t_{0})=0\), that is, there exist a pair of positive constants \((s_{0},t_{0})\) such that \(s_{0}u^{+}+t_{0}u^{-}\in \mathcal{M}\).
Next, we prove the uniqueness of \((s_{0},t_{0})\). Proofing by contradiction, we suppose that there are two unequal pairs of positive constants \((s_{1},t_{1})\) and \((s_{2},t_{2})\) such that \(s_{1}u^{+}+t_{1}u^{-}\in \mathcal{M}\) and \(s_{2}u^{+}+t_{2}u^{-}\in \mathcal{M}\). Note that the function \(f(x)=\frac{1-a^{x}}{x}\) is strictly monotonically decreasing on \((0,+\infty )\) for \(a>0\) and \(a\neq 1\). Hence, taking u, s, and t as \(s_{1}u^{+}+t_{1}u^{-}\), \(\frac{s_{2}}{s_{1}}\), and \(\frac{t_{2}}{t_{1}}\) in Lemma 2.2, respectively, and noting that \(p< q\), then we have
where \(\Theta ''= \frac{2(\frac{s_{2}}{s_{1}})^{p}C_{\frac{p}{2}-1}^{i}C_{i}^{j}+(\frac{s_{2}}{s_{1}})^{p}C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j} +2(\frac{t_{2}}{t_{1}})^{p}C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j-1}+(\frac{t_{2}}{t_{1}})^{p}C_{\frac{p}{2}-1}^{i-1}C_{i-1}^{j} -2(\frac{s_{2}}{s_{1}})^{p-(i+j)}(\frac{t_{2}}{t_{1}})^{i+j}C_{\frac{p}{2}}^{i}C_{i}^{j}}{2p} \geq 0\) (see Appendix 3). Also, taking u, s, and t as \(s_{2}u^{+}+t_{2}u^{-}\), \(\frac{s_{1}}{s_{2}}\), and \(\frac{t_{1}}{t_{2}}\), respectively, we have
As a consequence, there is a contradiction between (2.26) and (2.27). Hence, \((s_{1},t_{1})=(s_{2},t_{2})\) that implies that there is a unique pair of positive constants \((s_{0},t_{0})\) such that \(s_{0}u^{+}+t_{0}u^{-}\in \mathcal{M}\). □
Lemma 2.8
Assume that \((C_{1})\) and \((C_{2})\) hold. Then,
and
Proof
On the one hand, according to Corollary 2.4 and the definition of \(\mathcal{M}\), there holds
On the other hand, for any \(u\in \mathcal{D}\) with \(u^{\pm}\neq 0\), by virtue of Lemma 2.7 there appear two positive constants \(s_{0}\), \(t_{0}\) such that \(s_{0}u^{+}+t_{0}u^{-}\in \mathcal{M}\). Then, we have
which implies that
Hence, it is easy to see that the conclusion (2.29) holds. Similarly, it follows from Corollary 2.5, the definition of \(\mathcal{N}\), and Lemma 2.6 that (2.28) also holds. □
Lemma 2.9
Assume that \((C_{1})\) and \((C_{2})\) hold. Then, \(m_{*}>0\) and \(c_{*}>0\) can be achieved.
Proof
For any \(u\in \mathcal{M}\), there holds \(\langle I'(u),u\rangle =0\). For \(\varepsilon _{3}=\frac{b_{0}}{pc_{0}}>0\), by (1.6), (1.7), and Lemma 2.1, there is a positive constant \(C_{\varepsilon _{3}}\) such that
Since \(1< p< q<\zeta \), then \(\|u\|\geq \rho := ( \frac{(p-1)b_{0}^{\frac{\zeta}{p}}}{pc_{0}C_{\varepsilon _{3}}} )^{\zeta -p}\) for any \(u\in \mathcal{M}\).
Let \(\{u_{k}\}\subset \mathcal{M}\) be such that \(I(u_{k})\rightarrow m_{*}\). By (1.4) and (1.7), there holds
This shows that the sequence \(\{u_{k}\}\) is bounded in \(\mathcal{D}\), that is, there exists a \(M_{1}>0\) such that \(\|u_{k}\|\leq M_{1}\). Thus, there appears a \(u_{0}\in \mathcal{D}\) such that \(u_{k}^{\pm}\rightharpoonup u_{0}^{\pm}\) in \(\mathcal{D}\). Then, according to Lemma 2.1, we can obtain that \(u_{k}^{\pm}\rightarrow u_{0}^{\pm}\) in \(l^{\kappa}(\mathbb{Z},\mathbb{R})\) for \(\kappa \in [p,+\infty ]\) and \(u_{k}^{\pm}(n)\rightarrow u_{0}^{\pm}(n)\) for all \(n\in \mathbb{Z}\).
Since \(\{u_{k}\}\subset \mathcal{M}\), there exists \(\langle I'(u_{k}),u_{k}^{\pm}\rangle =0\) and then by Proposition 2.1, we have
and
It follows from (1.6), (2.30), Lemma 2.1, and the boundedness of \(\{u_{k}\}\) that there exists \(\varepsilon _{4}\in (0,\frac{b_{0}\rho ^{p}}{c_{0}M_{1}^{p}})\) and a positive constant \(C_{\varepsilon _{4}}\) such that
which implies that \(\|u_{k}^{+}\|_{l^{\zeta}}^{\zeta}\geq \frac{\rho ^{p}-c_{0}\varepsilon _{4}b_{0}^{-1}M_{1}^{p}}{c_{0}C_{\varepsilon _{4}}}>0\). Similarly, by (1.6), (2.31), Lemma 2.1, and the boundedness of \(\{u_{k}\}\), there exists \(\varepsilon _{5}\in (0,\frac{b_{0}\rho ^{p}}{c_{0}M_{1}^{p}})\) and a positive constant \(C_{\varepsilon _{5}}\) such that \(\|u_{k}^{-}\|_{l^{\zeta}}^{\zeta}\geq \frac{\rho ^{p}-c_{0}\varepsilon _{5} b_{0}^{-1}M_{1}^{p}}{c_{0}C_{\varepsilon _{5}}}>0\). Then, let \(\varepsilon ':=\max \{\varepsilon _{4},\varepsilon _{5}\}\) and \(C_{\varepsilon '}:=\max \{C_{\varepsilon _{4}},C_{\varepsilon _{5}} \}\), we have that \(\|u_{k}^{\pm}\|_{l^{\zeta}}^{\zeta}\geq \frac{\rho ^{p}-c_{0}\varepsilon ' b_{0}^{-1}M_{1}^{p}}{c_{0}C_{\varepsilon '}}>0\). For any \(p\leq \kappa \leq +\infty \), by virtue of the compactness of the embedding \(\mathcal{D}\hookrightarrow l^{\kappa}(\mathbb{Z},\mathbb{R})\) and
which implies that \(u_{0}^{\pm}\neq 0\). Note that \(\Delta u^{\pm}(n)=u^{\pm}(n+1)-u^{\pm}(n)\). By the fact that \(u_{k}^{\pm}(n)\rightarrow u_{0}^{\pm}(n)\) for all \(n\in \mathbb{Z}\), we can derive that
which implies that \(\Delta u^{\pm}_{k}(n)\rightarrow \Delta u^{\pm}_{0}(n)\) for all \(n\in \mathbb{Z}\). Note that
Also, by (1.6), for any given \(\varepsilon >0\), there exists \(C_{\varepsilon}>0\) such that
Then, we can obtain that \(|u_{k}^{+}(n)|^{q}|\ln |u_{k}^{+}(n)|^{r}|\leq \varepsilon |u_{k}^{+}(n)|^{p}+C_{ \varepsilon }|u_{k}^{+}(n)|^{\zeta}\leq \varepsilon b_{0}^{-1}M_{1}^{p}+ C_{ \varepsilon }b_{0}^{-\frac{\zeta}{p}}M_{1}^{\zeta}\), which implies that
Note that \(\sum_{n\in \mathbb{Z}}c(n)<\infty \) (by \((C_{2})\)). Thus, it follows from (2.30), the weak lower semicontinuity of norm, Fatou’s Lemma, and the Lebesgue dominated convergence theorem that
which implies that
Similarly, by (2.31), the weak lower semicontinuity of norm, Fatou’s Lemma, and the Lebesgue dominated convergence theorem, there exists
According to Lemma 2.7, there are two positive constants \(s_{3}\), \(t_{3}\) such that
By (1.4), (1.7), the weak lower semicontinuity of norm, \((C_{2})\), (2.32), Lemma 2.2, (2.34), (2.35), and (2.36), there exists
Moreover, in combination (2.37) with (2.35), we can obtain that
which implies that \(\langle I'(u_{0}),u_{0}^{+} \rangle \geq 0\). Similarly, we can also obtain that \(\langle I'(u_{0}),u_{0}^{-} \rangle \geq 0\). Then, by (2.34) and (2.35), we have \(\langle I'(u_{0}),u_{0}^{\pm }\rangle =0\) and then \(\langle I'(u_{0}),u_{0} \rangle =0\). Furthermore, according to (2.37), we can obtain that \(I(u_{0})=m_{*}\) and \(u_{0}\in \mathcal{M}\). Note that \(u_{0}^{+}\neq 0\). If we let \(s_{3}=0\) and \(t_{3}=0\) in (2.11), then we have
Through arguments similar to the above, we can also conclude that \(c_{*}>0\) can be achieved. □
Lemma 2.10
Assume that \((C_{1})\) and \((C_{2})\) hold. If \(u_{0}\in \mathcal{M}\) and \(I(u_{0})=m_{*}\), then \(u_{0}\) is a critical point of I.
Proof
Arguing by contradiction. If we suppose that \(I'(u_{0})\neq 0\) for all \(u_{0}\in \mathcal{D}\), then there are two positive constants δ and ϑ such that
Since \(u_{0}\in \mathcal{M}\), we have \(\langle I'(u_{0}),u_{0}^{\pm}\rangle =0\), and by Lemma 2.2, for all \(s,t\geq 0\), there exists
Let \(D=(\frac{1}{2},\frac{3}{2})\times (\frac{1}{2},\frac{3}{2})\). It follows from (2.38) that
For \(\epsilon :=\min \{\frac{m_{*}-\mathcal{Y}}{3}, \frac{\vartheta \delta}{8}\}\) and \(S_{\delta}:=B(u_{0},\delta )\), by [22], we can obtain a deformation \(\eta \in \mathcal{C}([0,1]\times \mathcal{D},\mathcal{D})\) such that
-
(i)
\(\eta (1,u)=u\) if \(|I(u)-m_{*}|>2\epsilon \);
-
(ii)
\(\eta (1,I^{m_{*}+\epsilon}\cap S_{\delta})\subset I^{m_{*}- \epsilon}\), where \(I^{c}:=\{u\in \mathcal{D}:I(u)\leq c\}\);
-
(iii)
\(I(\eta (1,u))\leq I(u)\), \(\forall u\in{\mathcal{D}}\);
-
(iv)
\(\eta (1,u)\) is a homeomorphism of \(\mathcal{D}\).
By virtue of (2.38), (iii), and for all \(s,t\geq 0\), which makes \(|s-1|^{2}+|t-1|^{2}\geq \delta ^{2}/{\|u_{0}\|^{2}}\) hold, there exists
Also, by Corollary 2.4, for all \(s,t\geq 0\), we have that \(I(su_{0}^{+}+tu_{0}^{-})\leq I(u_{0})=m_{*}\). Then, by (ii), we have
By virtue of (2.39), (2.40), and (2.41), we have
Define \(k(s,t)=su_{0}^{+}+tu_{0}^{-}\). Next, we prove that \(\eta (1,k(D))\cap \mathcal{M}\neq \varnothing \). Set \(\gamma (s,t):=\eta (1,k(s,t))\),
and
Note that \(\phi _{1}(s,t)\) and \(\phi _{2}(s,t)\) are two-dimensional vectors. According to (2.21) and (2.22), it is obvious that \(y_{1}(s,t)=\frac{1}{s}h_{1}(s,t)\) and \(y_{2}(s,t)=\frac{1}{t}h_{2}(s,t)\). Hence, \(\phi _{1}(s,t)\) is a \(C^{1}\) function of s, t and we have
Similarly, we also have
Let
and
where
and
It follows from \(1< p< q\) and \(u_{0}^{\pm}\neq 0\) that \(C^{+}<0\), \(C^{-}<0\), \(\frac{\partial y_{1}(s,t)}{\partial t}|_{(1,1)}\geq 0\), and \(\frac{\partial y_{2}(s,t)}{\partial s}|_{(1,1)}\geq 0\). Then, we have
which implies that \(\det M\neq 0\). According to the topological degree theory [1], we can obtain that \(\operatorname{deg}(\phi _{1},D,(0,0))=1\). By virtue of (2.39) and (i), there exists \(\gamma =k\) on ∂D. As a consequence, it follows from the homotopy invariance of the Brouwer degree that
which implies that \(\phi _{2}(s_{4},t_{4})=0\) for some \((s_{4},t_{4})\in D\) and so \(\eta (1,k(s_{4},t_{4}))=\gamma (s_{4},t_{4})\in \mathcal{M}\). Then, by the definition of \(\mathcal{M}\), we know that \(I(\eta (1,k(s_{4},t_{4})))\ge m^{*}\). This contradicts (2.42). Hence, \(I'(u_{0})=0\), that is, \(u_{0}\) is a critical point of I. □
3 The existence of sign-changing solutions
In this section, we will prove the existence of sign-changing solutions that only change sign once.
Proof of Theorem 1.1
First, it follows from Lemma 2.9 and Lemma 2.10 that problem (1.2) has a sign-changing solution \(u_{0}\in \mathcal{M}\) such that
Next, we prove that \(u_{0}\) only changes sign once. Denote \(u_{0}=u_{1}+u_{2}+u_{3}\), where
and \(V_{1}=\{n_{1},n_{1}+1,\ldots ,n_{1}+m_{1}\}\), \(V_{2}=\{n_{2},n_{2}+1,\ldots ,n_{2}+m_{2}\}\), where the value of \(n_{1}\) or \(n_{2}\) may be −∞ and the value of \(n_{1}+m_{1}\) or \(n_{2}+m_{2}\) may be +∞.
Setting \(w=u_{1}+u_{2}\), it is easy to see that \(w^{+}=u_{1}\), \(w^{-}=u_{2}\), and \(w^{\pm}\neq 0\). According to Lemma 2.7, there is a unique pair of positive constants \(s_{4}\), \(t_{4}\) such that \(s_{4}w^{+}+t_{4}w^{-}\in \mathcal{M}\). By virtue of \(I'(u_{0})=0\), we can derive that \(\langle I'(u_{0}),w^{\pm}\rangle =0\). Then, by (1.7), we can obtain that
Note that
According to (3.3), one has
Similarly, we can obtain that
On the basis of (1.4), (1.7), (2.11), (3.1), (3.2), (3.4), and (3.5), using the same processing method as (2.7), we have
which implies that \(u_{3}=0\). Thus, \(u_{0}\) only changes sign once. □
4 The existence of ground-state solutions
In this section, we will prove the existence of Nehari-type ground-state solutions for (1.2) and provide the relationship between the sign-changing ground-state energy and the ground-state energy. We mainly use the method in [2, 4] to prove that the functional I satisfies the Cerami condition at any level \(d\in (0,\infty )\), and then use the method in [5] to prove that the functional I has a mountain-pass geometry. To prove the above conclusions, we need the following lemmas.
Lemma 4.1
([18])
Let X be a real Banach space. For some constants \(\alpha ,\beta ,\rho >0\), and \(e\in X\) with \(\|e\|_{X}>\rho \), there exists a functional \(I\in C^{1}(X,\mathbb{R})\) satisfying the following mountain-pass geometry:
Set \(d_{0}=\inf_{\gamma \in \Gamma}\max_{0\leq t\leq 1} I (\gamma (t) )\), where \(\Gamma = \{\gamma \in C([0,1],X):\gamma (0)=0 \textit{ and } \gamma (1)=e \}\). Then, there exists a Cerami sequence \(\{u_{k}\}\subset X\) of I at level \(d_{0}\), where a sequence \(\{u_{k}\}\) is called a Cerami sequence at a level \(d_{0}\) if it satisfies
Remark 4.1
It is easy to obtain that \(d_{0}\ge \beta >0\) (for example, see the proof of Theorem 1.15 in [22]).
Lemma 4.2
The Cerami sequence \(\{u_{k}\}\subset \mathcal{D}\) at any level \(d_{0}\in (0,+\infty )\) has at least one convergent subsequence in \(\mathcal{D}\).
Proof
Since \(\{u_{k}\}\) is a Cerami sequence at the level \(d_{0}\), then (4.1) holds. We claim that \(\{u_{k}\}\) is bounded in \(\mathcal{D}\). Arguing by contradiction, we suppose that \(\{u_{k}\}\) is not bounded in \(\mathcal{D}\), that is, there appears a subsequence, still denoted by \(\{u_{k}\}\), such that \(\|u_{k}\|\rightarrow +\infty \) as \(k\rightarrow +\infty \).
Let \(w_{k}=\frac{u_{k}}{\|u_{k}\|}\). Then, there exists a subsequence, still denoted by \(\{w_{k}\}\), and a function \(w\in \mathcal{D}\) such that
Then, we will prove the claim by discussing the following two cases.
Case 1: \(w=0\).
Set \(t_{k}\in [0,1]\) such that \(I(t_{k}u_{k})=\max_{t\in [0,1]} I(tu_{k})\). For any given constants \(\tau >0\) and \(N>0\), it follows from the unboundedness of \(\{u_{k}\}\) that
Set \(\bar{w}_{k}=(p\tau +1)^{\frac{1}{p}}w_{k}\). By virtue of \((C_{2})\), the boundedness of \(\{w_{k}\}\), (2.1), (1.6), and the Lebesgue dominate convergence theorem, we have
Then, for k large enough, we can obtain that
According to the arbitrariness of τ, we can obtain that
If \(t_{k}=1\), substituting it into (4.3) can obtain \(\lim_{k\rightarrow \infty} I(u_{k})=+\infty \), which contradicts (4.1). Then, it follows from \(I(0)=0\) that \(t_{k}\in (0,1)\). Thus, \(\frac{d}{dt} I(tu_{k})\mid _{t=t_{k}}=0\). Therefore, according to the definition of \(\{u_{k}\}\), we can obtain that
which is contrary to (4.3). Hence, the assumption is not valid, that is, \(\{u_{k}\}\) is bounded in \(\mathcal{D}\).
Case 2: \(w\neq 0\).
Let \(V'= \{n\in \mathbb{Z}; w\neq 0 \}\). Then, \(|u_{k}(n)|\to +\infty \) as \(k\rightarrow +\infty \) for each \(n\in V'\). According to the fact that \(\|u_{k}\|\to +\infty \), as \(k\to +\infty \), and \(I(u_{k})\leq c_{*}\), there holds \(\frac{I(u_{k})}{\|u_{k}\|^{p}}\to 0\), as \(k\to +\infty \), that is
which together with the definition of \(\mathcal{D}\), \((C_{1})\), and \((C_{2})\) implies that \(G(u_{k}):= \frac{r\sum_{n\in \mathbb{Z}}c(n)|u_{k}(n)|^{q}}{q^{2}\|u_{k}\|^{p}} - \frac{\sum_{n\in \mathbb{Z}}c(n)|u_{k}(n)|^{q}\ln{|u_{k}(n)|^{r}}}{q\|u_{k}\|^{p}}\) is bounded, that is, there is a positive constant \(M_{2}\) such that
We set
where \(M_{3}=e^{\frac{1}{q}}>0\) and \((\cdot )=\frac{rc(n)|u_{k}(n)|^{q}}{q^{2}\|u_{k}\|^{p}} - \frac{c(n)|u_{k}(n)|^{q}\ln{|u_{k}(n)|^{r}}}{q\|u_{k}\|^{p}}\). For I in (4.5), according to (1.6), Lemma 2.1, (2.33), and \(\sum_{n\in \mathbb{Z}}c(n)<+\infty \), there are two positive constants \(\varepsilon _{6}\) and \(C_{\varepsilon _{6}}\) such that
For II in (4.5), we have
Note that \(|u_{k}(n)|\to +\infty \) as \(k\rightarrow +\infty \) for each \(n\in V'\). Then, similar to the argument of II, we also have
Thus, \(\lim_{k\to \infty}G(u_{k})=-\infty \), which contradicts (4.4). Therefore, we deduce that \(\{u_{k}\}\) is bounded in \(\mathcal{D}\).
As a consequence, both of the above cases indicate that the assumption is not valid, that is, \(\{u_{k}\}\) is bounded in \(\mathcal{D}\). Then, there exists a subsequence, still denoted by \(\{u_{k}\}\), and a function \(u\in \mathcal{D}\) such that
Note that \(\{u_{k}\}\) is a Cerami sequence. Then, there holds
Moreover, by (4.9), we have
Note that \(\{u_{k}\}\) is bounded in E. On the basis of (1.6), (2.1), (4.9), and \(\sum_{n\in \mathbb{Z}}c(n)<+\infty \) there exist two positive constants ε and \(C_{\varepsilon}\) such that
Similarly, it follows from the boundedness of \(\{\|u_{k}\|\}\), (1.6), (2.1), (4.9), and \(\sum_{n\in \mathbb{Z}}c(n)<+\infty \) that
Then, using the Hölder inequality
where \(d_{1}\), \(d_{2}\), \(d_{3}\), \(d_{4}\) are nonnegative constants and \(p^{*}=\frac{p}{p-1}\), \(p>1\), by virtue of (1.7) and (1.3), we have
According to (4.10), (4.11), (4.12), (4.13), and (4.14), we have \(\|u_{k}\|\to \|u\|\) as \(k\to +\infty \). By the uniform convexity of \(\mathcal{D}\) (similar to the argument of the Appendix A.1 in [24]), the fact that \(u_{k}\rightharpoonup u\) in \(\mathcal{D}\) and the Kadec–Klee property, we can obtain that \(u_{k}\to u\) in \(\mathcal{D}\). Thus, I satisfies the Cerami condition. □
Next, we prove that the functional I defined by (1.4) has a mountain-pass geometry.
Lemma 4.3
(i) There are two positive constants ρ and \(\delta '\) such that \(I(u)\geq \delta '\) for all \(u\in \mathcal{D}\) with \(\|u\|=\rho \).
(ii) There is \(\varphi _{j}\in \mathcal{D}\setminus \{0\}\) such that \(I(t\varphi _{j})\rightarrow -\infty \) as \(t\rightarrow +\infty \).
Proof
For (i), it follows from (1.4), (1.6), and Lemma 2.1 that there exists \(\varepsilon _{7}\in (0,\frac{qb_{0}}{pc_{0}})\) and \(C_{\varepsilon _{7}}>0\) such that
Choose \(\rho >0\) sufficiently small. There appears a constants \(\beta _{0}= \frac{\rho ^{p} q-\rho ^{p} pc_{0} (\varepsilon _{7}b_{0}^{-1}+C_{\varepsilon _{7}}b_{0}^{-\frac{\zeta}{p}} )}{pq}>0\) such that \(I(u)\geq \beta _{0}\) for all \(u\in \mathcal{D}\) with \(\|u\|=\rho \).
By the definition of \(c_{*}\), for any \(j>0\), we can choose a \(\varphi _{j}\in \mathcal{N}\subset \mathcal{D}\setminus \{0\}\) such that
Then, for any \(t>0\), there holds
which implies that \(I(t\varphi _{j})\rightarrow -\infty \) as \(t\rightarrow +\infty \), and hence, (ii) holds. □
Proof of Theorem 1.2
Lemma 4.1 and Lemma 4.3 imply that I has a Cerami sequence \(\{u_{kj}\}\) at the level \(d_{j}\), that is,
By virtue of Remark 4.1 and the definition of \(d_{j}\), it is easy to see that \(d_{j}\in [\beta _{0}, \max_{0\leq t\leq 1} I(t\varphi _{j})]\). Furthermore, noting that \(\varphi _{j}\in \mathcal{N}\), according to Corollary 2.5, we obtain that \(I(\varphi _{j})=\max_{t\ge 0} I(t\varphi _{j})\), and hence, \(d_{j}\in [\beta _{0}, I(\varphi _{j})]\), which together with (4.15) implies that \(d_{j}\in [\beta _{0}, c_{*}+\frac{1}{j}]\). Thus, we can choose a subsequence \(\{u_{k_{j},j}\}\), denoted by \(\{u_{j}\}\), such that
for some \(d_{*}\in [\frac{\beta _{0}}{2}, c_{*}]\). Equation (4.16) and Lemma 4.2 imply that \(\{u_{j}\}\) has a convergent subsequence, still denoted by \(\{u_{j}\}\), such that \(u_{j}\to \bar{u}\) as \(j\to +\infty \). By the continuity of I and \(I'\), we obtain that \(I(\bar{u})=d_{*}\) and \(I'(\bar{u})=0\), which together with the fact that \(d_{*}\ge \frac{\beta _{0}}{2}>0\) implies that \(\bar{u}\in \mathcal{N}\) is a nontrivial solution of (1.2) and obviously, \(I(\bar{u})\geq c_{*}=\inf_{u\in \mathcal{N}}I(u)\). Moreover, according to (4.16), (1.4), (1.7), and the weak lower semicontinuity of norm, there exists
which implies that \(I(\bar{u})\leq c_{*}\). Thus, \(I(\bar{u})=c_{*}=\inf_{u\in \mathcal{N}} I(u)>0\).
Finally, we prove that \(m_{*}\ge 2c_{*}\). In fact, it follows from Corollary 2.4, Proposition 2.1, Lemma 2.6, and Lemma 2.8 that there are two positive constants \(s'\) and \(t'\) such that \(s'u_{0}^{+}\in \mathcal{N}\) and \(t'u_{0}^{-}\in \mathcal{N}\), then one has
The proof of Theorem 1.2 is completed. □
Data Availability
No datasets were generated or analysed during the current study.
References
Chang, K.C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)
Chang, X.J., Wang, R., Yan, D.K.: Ground states for logarithmic Schrödinger equations on locally finite graphs. J. Geom. Anal. 33, 211 (2023)
Chen, G.W., Ma, S.W.: Discrete nonlinear Schrödinger equations with superlinear nonlinearities. Appl. Math. Comput. 218, 5496–5507 (2012)
Chen, P., Tang, X.H.: Infinitely many homoclinic solutions for the second-order discrete p-Laplacian systems. Bull. Belg. Math. Soc. Simon Stevin 20, 193–212 (2013)
Chen, S.T., Tang, X.H., Yu, J.S.: Sign-changing ground state solutions for discrete nonlinear Schödinger equations. J. Differ. Equ. Appl. 25, 202–218 (2019)
Cheng, B.T., Chen, J.H., Zhang, B.L.: Least energy nodal solution for Kirchhoff-type Laplacian problems. Math. Methods Appl. Sci. 43, 3827–3849 (2020)
Christodoulides, D.N., Lederer, F., Silberberg, Y.: Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003)
Flach, S., Gorbach, A.V.: Discrete breathers-advances in theory and applications. Phys. Rep. 467, 1–116 (2008)
Flach, S., Willis, C.R.: Discrete breathers. Phys. Rep. 295, 181–264 (1998)
Heenig, D., Tsironis, G.P.: Wave transmission in nonlinear lattices. Phys. Rep. 307, 333–432 (1999)
Jia, L.Q., Chen, G.W.: Discrete Schrödinger equations with sign-changing nonlinearities: infinitely many homoclinic solutions. J. Math. Anal. Appl. 452, 568–577 (2017)
Kulpa, W.: The Poincaré-Miranda theorem. Am. Math. Mon. 104, 545–550 (1997)
Liu, X., Shi, H.P., Zhang, Y.B.: Existence of periodic solutions of second order nonlinear p-Laplacian difference equations. Acta Math. Hung. 133, 148–165 (2011)
Mai, A., Zhou, Z.: Discrete solitons for periodic discrete nonlinear Schrödinger equations. Appl. Math. Comput. 222, 34–41 (2013)
Mei, P., Zhan, Z., Chen, Y.M.: Homoclinic solutions of discrete p-Laplacian equations containing both advance and retardation. Electron. Res. Arch. 30, 2205–2219 (2022)
Niculescu, C.P., Stǎanescu, M.M.: A note on Abel’s partial summation formula. Aequ. Math. 91, 1009–1024 (2017)
Ou, X., Zhang, X.Y.: Ground state sign-changing solutions for second order elliptic equation with logarithmic nonlinearity on locally finite graphs. arXiv:2306.10302v1
Schechter, M.: A variation of the mountain pass lemma and applications. J. Lond. Math. Soc. 44, 491–502 (1991)
Shi, H.P., Liu, X., Zhang, Y.B.: Existence of periodic solutions of 2nth-order nonlinear p-Laplacian difference equations. Rocky Mt. J. Math. 46, 1679–1699 (2016)
Shi, H.P., Zhang, Y.B.: Stangding wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials. Appl. Math. Lett. 58, 95–102 (2016)
Tang, X.H., Cheng, B.T.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261, 2384–2402 (2016)
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
Yang, M.B., Chen, W.X., Ding, Y.H.: Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta Appl. Math. 110, 1475–1488 (2010)
Yang, P., Zhang, X.Y.: Existence and multiplicity of nontrivial solutions for a \((p,q)\)-Laplacian system on locally finite graphs. arXiv:2304.12676
Funding
This project is supported by Yunnan Fundamental Research Projects (grant No: 202301AT070465) and supported by the Yunnan Ten Thousand Talents Plan Young & Elite Talents Project.
Author information
Authors and Affiliations
Contributions
Ou and Zhang contributed equally to this work. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendices
Appendix 1
There exists \(u\in E\) such that \(\sum_{n\in \mathbb{Z}}c(n)|u(n)|^{q}\ln{|u(n)|^{r}}=- \infty \).
Proof
Set
where \(|n|\) represents the absolute value of n. Let
Using the method for discriminating the convergence of improper integrals, we can obtain the following results
According to the definitions of Δ, \(u(n)\) and \(a(n)\), using the \(C_{p}\) inequality, we have
Then, according to the definition of \(u(n)\) and \(b(n)\), there holds
Therefore, it is easy to see that \(u\in E\).
Now, we prove that \(\sum_{n\in \mathbb{Z}}c(n)|u(n)|^{q}\ln{|u(n)|^{r}}=-\infty \) if \(1< q\leq 2\). Note that if \(|n|\ge p+2\), then
Thus, there holds
As \(|n|\geq p+2\), we have \(\ln \ln |n|>0\) and then \(II'>0\). On the other hand, it is easy to see that \(I'=\infty \) since \(1< q\leq 2\). Thus, we complete the proof. □
Appendix 2
For all \(s,t\geq 0\) and \(0\leq i,j\leq \frac{p}{2}\), there exist
Proof
Using the combination formula \(C_{p-1}^{j}=\frac{p-j}{p}C_{p}^{j}\), \(C_{p-1}^{j-1}=\frac{j}{p}C_{p}^{j}\) and deformation of the Young inequality \(x^{\lambda }y^{1-\lambda}\leq \lambda x+(1-\lambda )y\), \((0<\lambda <1)\), there exists
Similarly, we can obtain that \(\frac{2s^{p}C_{\frac{p}{2}-1}^{i}+s^{p}C_{\frac{p}{2}-1}^{i-1}+t^{p}C_{\frac{p}{2}-1}^{i-1}-2s^{p-i}t^{i}C_{\frac{p}{2}}^{i}}{2p} \geq 0\), \(\frac{s^{p} C_{\frac{p}{2}-1}^{i}+t^{p}C_{\frac{p}{2}-1}^{i-1}-s^{p-2i}t^{2i}C_{\frac{p}{2}}^{i}}{p} \geq 0\) and \(\frac{s^{p}C_{\frac{p}{2}-1}^{j}+2t^{p}C_{\frac{p}{2}-1}^{j-1}+t^{p}C_{\frac{p}{2}-1}^{j} -2s^{\frac{p}{2}-j}t^{\frac{p}{2}+j}C_{\frac{p}{2}}^{j}}{2p}\geq 0\). Thus, we can see that the conclusions hold. □
Appendix 3
For all \(\frac{s_{2}}{s_{1}},\frac{t_{2}}{t_{1}}\geq 0\) and \(0\leq i,j\leq \frac{p}{2}\), there exists
Proof
Using the method of Appendix 2, we have
Thus, we complete the proof. □
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ou, X., Zhang, X. Ground-state sign-changing homoclinic solutions for a discrete nonlinear p-Laplacian equation with logarithmic nonlinearity. Bound Value Probl 2024, 6 (2024). https://doi.org/10.1186/s13661-023-01811-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-023-01811-5