Ground state sign-changing homoclinic solutions for a discrete nonlinear $p$-Laplacian equation with logarithmic nonlinearity

By using a direct non-Nehari manifold method from [X.H. Tang, B.T. Cheng. J. Differ. Equations. 261(2016), 2384-2402.], we obtain an existence result of ground state sign-changing homoclinic solution which only changes sign one times and ground state homoclinic solution for a class of discrete nonlinear $p$-Laplacian equation with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground state energy is larger than twice of the ground state energy.


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: ⎧ ⎨ ⎩ (ϕ p ( u(n -1)))a(n)ϕ p (u(n)) + ∇W (n, u(n)) = 0, n ∈ Z, lim n→±∞ u(n) = 0, (1.1) where p > 1, ϕ p is the p-Laplace operator, u ∈ R N , a : Z → R and W : Z × R N → 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][8][9][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 groundstate 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 → R. When a is bounded from below and the volume of set {x ∈ V : a(x) ≤ M} is finite, they used the Nehari manifold method to obtain that the equation has a groundstate solution.Moreover, when a is bounded from below and 1/a(x) is a Lebesgue integrable function on the set {x ∈ 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 is a bounded domain in 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, ϕ p (s) = |s| p-2 s is the p-Laplacian operator, p 2 ∈ N * , N * denotes the positive integer set, a, b, c : Z → (0, +∞), r ≥ 1, u : Z → R, and 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 D, thereby avoiding the case that n∈Z c(n)|u(n)| q ln |u(n)| r = -∞.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) ≥ b 0 for all n ∈ Z and lim |n|→+∞ b(n) = +∞; (C 2 ) there is a positive constant c 0 such that c(n) ≤ c 0 for all n ∈ Z and n∈Z c(n) < +∞.Next, we define Then, E is a reflexive Banach space.As usual, let 1 < p < +∞ and define with the norm .
When p = +∞, we define Note that equation (1.2) is formally related to the energy functional I : E → R ∪ {+∞} 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 lim t→0 t q-1 ln |t| r t p-1 = 0 and lim for all n ∈ Z, where ζ ∈ (q, +∞).Then, by (C 2 ), for any given ε > 0, there exists a positive constant Then, D is the closed subspace of E, I ∈ C 1 (D, R) and Using Abel's partial summation formula (also known as Abel's transformation) in [16] and the definition of u(n), we have which implies that According to the above equations, we can derive that I (u), v = 0 for any v ∈ D if and only if Therefore, it is easy to see that the critical points of I in D are solutions of equation (1.2).Furthermore, if u ∈ D is a solution of equation (1.2) and u ± = 0, then u is a sign-changing solution of equation (1.2), where To be precise, we obtain the following results.In addition, m * ≥ 2c * .

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, D is continuously embedded into l κ (Z, R) for any p ≤ κ ≤ +∞, that is, for all u ∈ D, Moreover, D is compactly embedded in l κ (Z, R) for any p ≤ κ ≤ +∞.
Next, we prove that the embeddings are also compact.Suppose that {u k } is a bounded sequence in D.Then, there is a subsequence of {u k }, still denoted by {u k }, such that u k u weakly in D for some point u ∈ D. In particular, where ϕ ∈ D is defined by for any fixed n.Thus, we have We now prove u k → u in l κ (Z, R) for all p ≤ κ ≤ +∞.When κ = p, since u ∈ D, according to the boundedness of {u k } and the definition of • , there appears a positive constant δ 0 such that For any given positive constant ε 1 , there is a n 0 ∈ Z such that 1 b(n) < ε 1 as |n| > n 0 .Therefore, we can obtain that On the other hand, (2.3) implies that Then, according to the arbitrariness of ε 1 and (2.4), we have For κ = +∞, according to the definition of • l ∞ and (2.5), as k → +∞, we have and for p < κ < +∞, by (2.5) and (2.6), there exists Consequently, by (2.5), (2.6), and (2.7), we can derive that u k → u in l κ (Z, R) for all p ≤ κ ≤ +∞.
Proposition 2.1 For all p 2 ∈ N * and u ∈ D, there hold Then, according to the definition of • , Appendix 1 below, and the binomial theorem, we have Similarly, we have (2.10) By (1.4), (1.7), (2.8), (2.9), and (2.10), it is easy to see that the conclusions hold.
Note that 1 < p < q, ≥ 0 and the function f (x) = 1-a x x is strictly monotonically decreasing on (0, +∞) for a > 0 and a = 1.Then, in combination with Lemma 2.2, we have the following corollary.
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 ∈ N , there holds I(u) = max t≥0 I(tu).Lemma 2.6 Assume that (C 1 ) and (C 2 ) hold.For any u ∈ D with u = 0, there exists a unique positive constant t 0 such that t 0 u ∈ N .
Proof First, we prove the existence of t 0 .For any u ∈ D with u = 0, let u ∈ N be fixed and define a function g(t) = I (tu), tu on (0, +∞).On the one hand, by (1.6) and Lemma 2.1, there exist two positive constants ε 2 < b 0 c 0 and C ε 2 such that (2.17) Then, according to ζ > 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 ∈ Z, by (C 2 ) and (1.6), there exists Then, by 1 < p < q, r ≥ 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 ∈ (0, +∞) such that g(t 0 ) = 0, which implies that there exists a positive constant t 0 such that t 0 u ∈ N .Next, we prove the uniqueness of t 0 .Proofing by contradiction, we assume that there exist u ∈ D and two positive numbers t 1 = t 2 such that t 1 u ∈ N and t 2 u ∈ N .Note that the function f (x) = 1-a x  x is strictly monotonically decreasing on (0, +∞) for a > 0 and a = 1.
Taking u as t 1 u and t as t 2 t 1 in Corollary 2.3, there holds On the other hand, taking u as t 2 u and t as 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 ∈ N .
Lemma 2.7 Assume that (C 1 ) and (C 2 ) hold.For any u ∈ D with u ± = 0, there exists a unique pair of positive constants (s 0 , t 0 ) such that s 0 u + + t 0 u -∈ 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 -∈ M and s 2 u + + t 2 u -∈ M. Note that the function f (x) = 1-a x  x is strictly monotonically decreasing on (0, +∞) for a > 0 and a = 1.Hence, taking u, s, and t as s 1 u + + t 1 u -, s 2 s 1 , and t 2 t 1 in Lemma 2.2, respectively, and noting that p < q, then we have where = 2( ≥ 0 (see Appendix 3).Also, taking u, s, and t as s 2 u + + t 2 u -, s 1 s 2 , and 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 -∈ M.
On the other hand, for any u ∈ D with u ± = 0, by virtue of Lemma 2.7 there appear two positive constants s 0 , t 0 such that s 0 u + + t 0 u -∈ M.Then, we have max s,t≥0 Hence, it is easy to see that the conclusion (2.29) holds.Similarly, it follows from Corollary 2.5, the definition of N , and Lemma 2.6 that (2.28) also holds.
Proof For any u ∈ M, there holds I (u), u = 0.For ε 3 = b 0 pc 0 > 0, by (1.6), (1.7), and Lemma 2.1, there is a positive constant C ε 3 such that (1.4) and (1.7), there holds This shows that the sequence {u k } is bounded in D, that is, there exists a M 1 > 0 such that u k ≤ M 1 .Thus, there appears a u 0 ∈ D such that u ± k u ± 0 in D.Then, according to Lemma 2.1, we can obtain that u Since {u k } ⊂ M, there exists I (u k ), u ± k = 0 and then by Proposition 2.1, we have (2.31) It follows from (1.6), (2.30), Lemma 2.1, and the boundedness of {u k } that there exists ) and a positive constant C ε 4 such that ) and a positive constant C ε 5 such that For any p ≤ κ ≤ +∞, by virtue of the compactness of the embedding D → l κ (Z, R) and (2.32) Also, by (1.6), for any given ε > 0, there exists C ε > 0 such that Then, we can obtain that Note that n∈Z c(n) < ∞ (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 which implies that I (u 0 ), u + 0 ≥ 0. Similarly, we can also obtain that I (u 0 ), u - 0 ≥ 0.Then, by (2.34) and (2.35), we have I (u 0 ), u ± 0 = 0 and then I (u 0 ), u 0 = 0. Furthermore, according to (2.37), we can obtain that I(u 0 ) = m * and u 0 ∈ M. Note that u + 0 = 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 ∈ M and I(u 0 ) = m * , then u 0 is a critical point of I.

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 ∈ M such that Next, we prove that u 0 only changes sign once.Denote u 0 = u 1 + u 2 + u 3 , where , 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 ± = 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 -∈ M. By virtue of I (u 0 ) = 0, we can derive that I (u 0 ), w ± = 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.

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 ∈ (0, ∞), 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 α, β, ρ > 0, and e ∈ X with e X > ρ, there exists a functional I ∈ C 1 (X, R) satisfying the following mountain-pass geometry: .Then, there exists a subsequence, still denoted by {w k }, and a function Then, we will prove the claim by discussing the following two cases.
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→∞ I(u k ) = +∞, which contradicts (4.1).
Then, it follows from I(0) = 0 that t k ∈ (0, 1).Thus, d dt I(tu k ) | 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, which together with the definition of D, (C 1 ), and (C 2 ) implies that G(u k ) := We set where For I in (4.5), according to (1.6), Lemma 2.1, (2.33), and n∈Z c(n) < +∞, there are two positive constants ε 6 and C ε 6 such that For II in (4.5), we have Note that |u k (n)| → +∞ as k → +∞ for each n ∈ V .Then, similar to the argument of II, we also have Thus, lim k→∞ G(u k ) = -∞, which contradicts (4.4).Therefore, we deduce that {u k } is bounded in D.
As a consequence, both of the above cases indicate that the assumption is not valid, that is, {u k } is bounded in D.Then, there exists a subsequence, still denoted by {u k }, and a function u ∈ D such that Note that {u k } is bounded in E. On the basis of (1.6), (2.1), (4.9), and n∈Z c(n) < +∞ there exist two positive constants ε and Similarly, it follows from the boundedness of { u k }, (1.6), (2.1), (4.9), and n∈Z c(n) < +∞ that According to (4.10), (4.11), (4.12), (4.13), and (4.14), we have u k → u as k → +∞.By the uniform convexity of D (similar to the argument of the Appendix A.1 in [24]), the fact that u k u in D and the Kadec-Klee property, we can obtain that u k → u in 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 δ such that I(u) ≥ δ for all u ∈ D with u = ρ.
As |n| ≥ p + 2, we have ln ln |n| > 0 and then II > 0. On the other hand, it is easy to see that I = ∞ since 1 < q ≤ 2. Thus, we complete the proof.
Appendix 2 For all s, t ≥ 0 and 0 ≤ i, j ≤ p 2 , there exist   Thus, we complete the proof.
of the Young inequality x λ y 1-λ ≤ λx + (1λ)y, (0 < λ < 1), there exists 2s Similarly, by (1.6), (2.31), Lemma 2.1, and the boundedness of {u k }, there exists ε 5 ∈ (0, b 0 ρ p Then, there exists a Cerami sequence {u k } ⊂ X of I at level d 0 , where a sequence {u k } is called a 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 D. Arguing by contradiction, we suppose that {u k } is not bounded in D, that is, there appears a subsequence, still denoted by {u k }, such that u k → +∞ as k → +∞.
[22]rk 4.1 It is easy to obtain that d 0 ≥ β > 0 (for example, see the proof of Theorem 1.15 in[22]).Lemma 4.2 The Cerami sequence {u k } ⊂ D at any level d 0 ∈ (0, +∞) has at least one convergent subsequence in D.