Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs

In this paper, we investigate a class of p-Laplacian systems on a locally finite graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document}. By exploiting the method of Nehari manifold and some new analytical techniques, under suitable assumptions on the potentials and nonlinear terms, we prove that the p-Laplacian system admits a ground state solution (uλ,vλ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_{\lambda },v_{\lambda })$$\end{document} when the parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is sufficiently large. Furthermore, we consider the concentration behavior of these solutions as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document}, and show that these solutions converge to a solution of the corresponding limit problem.


Introduction
Recently, the study of differential equations on discrete graphs has attracted much attention from many researchers, due to its strong application background, such as neural network [1], image processing [2], and so on. For example, there were some critical works given by Grigor'yan et al. [3][4][5]. They studied several nonlinear elliptic equations on graphs and first established the Sobolev spaces and the functional framework on graphs. Then the problem of solving the equations on graphs is transformed into finding critical points of various functionals. In addition, the solutions of the heat equation and its variations on graphs have also been investigated by many authors due to its wide range of applications ranging from modeling of energy flows through a network to image processing [6,7]. In particular, the existence and concentration behavior of the solutions of equations have been extensively studied on graphs; see for examples Zhang and Zhao [8], Han et al. [9,10]. However, as far as we know, this kind of result to the p-Laplacian system on graphs   Shao JFPTA has not been studied in the literature. The main concern of this article is to discuss the following p-Laplacian system on the locally finite graph G = (V, E), where Δ p is the discrete p-Laplacian on graphs, λ > 1 is a parameter, a and b are non-negative potential functions. F ∈ C 1 (R 2 , R + ) is a positively homogeneous function of degree γ and 2 ≤ p < γ < ∞.
In Euclidean space, the p-Laplacian Δ p u = div(|∇u| p−2 ∇u) arises in non-Newtonian fluids, flow through porous media, nonlinear elasticity, and other physical phenomena. Problems like (1) can be used to accurately describe the multiplicate chemical reaction catalyzed by the catalyst grains under constant or variant temperature, and a correspondence of the stable station of the dynamical system determined by the reaction-diffusion system. The existence of solutions to the system (1) has been extensively studied in Euclidean spaces. In the case of a single equation, Furtado [11] studied the following p-Laplacian equation and showed the existence of solutions which change sign exactly once and the concentration behavior of these solutions as λ → +∞. In the case of the system, Gheraibia [12] studied the existence, multiplicity, and asymptotic behavior of solutions for the strongly coupled quasilinear elliptic system with vanishing potentials. Especially, using the Nehari manifold method and the concentration compactness principle, Lü and Liu [13] established the existence and multiplicity results of ground state solution to a quasilinear Schrödinger system with homogeneous nonlinearities. For other related works on Euclidean space, we refer the reader to [14][15][16] and the references therein. On graphs, the discrete p-Laplacian was introduced in [17] and has been well studied ever since, mostly in the context of nonlinear potential and spectral theory, cf. [18,19] for historical overviews. Zhang and Zhao [8] first studied this kind of results to the second-order equation on a locally finite graph and proved that, as λ → +∞, the ground state solutions of (3) converge to a ground state solution of a corresponding Dirichlet equation. Later, results in [8] were generalized by Han and Shao [9] to a nonlinear p-Laplacian (p ≥ 2) equation on a locally finite graph with a more general nonlinear term. Furthermore, they proved the completeness, reflexivity, and other vital properties of the Sobolev Space W 1,p (V )(p ≥ 2) in [9], which are fundamental when dealing with equations on graphs under the variational framework. Quite recently, using the mountain pass theorem, Xu and Zhao [20] considered the elliptic systems on graphs when p = 2 and (1). They proved the existence and asymptotic behavior of the ground state solutions for the systems. Several interesting existence and uniqueness results on graphs can be found in [21][22][23][24][25]. Inspired by the work mentioned above, this paper aims to study the system (1) with homogeneous nonlinearities on the locally finite graph G = (V, E). Roughly speaking, we first prove that the system (1) has a ground state solution (u λ , v λ ). Next, to discuss the concentration behavior of (u λ , v λ ) as λ → ∞, we introduce the Dirichlet problem which can be seen as a limit problem for (1) when λ goes to infinity. Importantly, we show that, as λ → ∞, these solutions of (1) converge in W 1,p (V ) × W 1,p (V ) to a ground state solution of (4).
To the best of our knowledge, system (1) has not been considered before on graphs. Especially, we generalize the results in [20] for p = 2 to any p ≥ 2 with more general nonlinear terms and the method we used in the current paper is different from theirs. Specifically, if p = 2 and F (u, v) = |u| α |v| β , α, β > 1, α+β = γ in the system (1), Xu and Zhao [20] proved similar results to ours using the mountain pass theorem and proving that the least energy level and the mountain pass level are the same. Note that it is easy to verify that the corresponding energy functional of the system (1) satisfies the (P S) condition when p = 2, while it is hard when p > 2. The main problem is that we could not prove . Therefore, the methods in [20] are not applicable in this article. To overcome this difficulty, we use the Nehari manifold method and some new analytical techniques to obtain our main results without using the (P S) condition.
On the other hand, our proofs on a graph are different from those in the Euclidean space. It is worth pointing out that the nonlinear partial differential equations on graphs are quite different from equations set in Euclidean spaces. Since the gradient of a function cannot be defined on a discrete graph, it makes some difficulties and limitations to the study of partial differential equations on graphs. For example, when considering the existence of solutions for p-Laplacian systems in the Euclidean spaces, the classical inequality in [26][27][28], i.e., for any x, y ∈ R N , N ≥ 1, plays an important role in proving the (P S) condition. However, the above inequality (5) do not hold on the locally finite graphs, which requires us to find new tools to deal with such problems. Fortunately, the nonlinear term of the system (1) is homogeneous of degree γ, which has some useful properties (see Remark 1.2). In particular, we can prove directly, which is the key to obtain the existence of solutions for the system (1).
Before stating our main results, we introduce some assumptions and notations. Let G = (V, E) be a graph, where V denotes the set of vertices and E denotes the set of edges. The distance d(x, y) of two vertices x, y ∈ V is defined by the minimal number of edges which connect these two vertices. For a subset Ω of V , if the distance d(x, y) is uniformly bounded from above for any x, y ∈ Ω, we call Ω a bounded domain in V . We shall remark that a bounded domain of a locally finite graph can contain only finite vertices. The boundary of Ω in V is defined by ∈ Ω : ∃x ∈ Ω such that xy ∈ E} and the interior of Ω is denoted by Ω • . Moreover, we denote Ω = Ω ∪ ∂Ω.
In this paper, we always assume that G, a, b, and F satisfy the following conditions. (G 1 ) G is a locally finite and connected graph and its measure μ( Our main results are stated as follows. Our first result is about the existence of the ground state solutions of the system (1). Theorem 1.1. Assume that (G 1 ), (G 2 ), (A 1 ), (A 2 ) and (F 1 ) hold. Then for any λ > 1 and p ≥ 2, the system (1) has a ground state solution (u λ , v λ ).
Next, we get the existence of a ground state solution for the Dirichlet problem (4).

Theorem 1.2.
Let Ω, Ω a , and Ω b be non-empty and bounded domains in V . Suppose that (G 1 ), (G 2 ) and (F 1 ) hold. Then for any p ≥ 2, the system (4) has a ground state solution.
Finally, we have the following concentration result. Theorem 1.3. Under the same assumptions as in Theorem 1.1, we have that, for any sequence λ n → ∞, up to a subsequence, the corresponding ground state solutions (u λn , v λn ) of (1) converge in W 1,p (V ) × W 1,p (V ) to a ground state solution of (4).
where c i is a non-negative and bounded potential function in V .
important assumption, which ensures the reflexivity of the Sobolev space W 1,p (V ) (see Corollary 5.8 in [9]).
Remark 5]) Using (F 1 ), then we have the following conclusions. where is used to overcome the lack of compactness on graphs. To be specific, we prove that the compact embedding theorem of This article is organized as follows. In Sect. 2, we introduce some concepts, notations, and known results which are helpful to prove the results of this paper. In Sect. 3, we prove the existence of ground state solutions to the system (1). In Sect. 4, we investigate the convergence behavior of the ground state solutions. In Sect. 5, we give some concluding remarks and future research prospects.

Preliminaries and functional settings
In this section, we present some preliminaries and basic functional settings. First, we give some definitions and notations on graphs.
For any function u : V → R and x ∈ V , the μ-Laplacian (or Laplacian for short) of u at x is defined by For brevity, we use Γ(u) for Γ(u, u) and sometimes we use ∇u∇v instead of Γ(u, v). The length of Γ(u) at x ∈ V is denoted by The p-Laplacian of u is defined by For where Let C c (V ) be a set of all functions with finite support, then W 1,p (V ) is the completion of C c (V ) under the norm · W 1,p (V ) (see Proposition 5.7 in [9]). For any function h(x) ≥ 0 and given λ > 1, we define a subspace of W 1,p (V ), which is also a reflexive Banach space, namely The corresponding Euler-Lagrange functional of the system (1) is given by We can deduce from hypotheses (A 1 ), (A 2 ) and (F 1 ) that the functional J λ ∈ C 1 (V, R) and It is well known that the weak solutions of the system (1) are the critical points of the functional J λ .
Vol. 25 (2023) Existence and convergence of solutions for p-Laplacian systemsPage 7 of 21 50 To prove the existence of ground state solutions to the system (1), we define the following Nehari manifold related to (1) as It is suitable to study the problem (4) in the Sobolev space X Ω and the functional related to (4) is Next, we introduce two formulas of integration by parts on locally finite graphs. The proofs of the following two lemmas can be found in [9] and we omit them here.

Lemma 2.2.
Let Ω ⊂ V be a bounded domain and assume that u ∈ W 1,p (Ω).

Then for any
Finally, we prove the Sobolev embedding theorems on the graphs. Lemma 2.3. Assume that λ > 1, (A 1 ) and (A 2 ) hold. Then X λ is continuously embedded into L q (V, R 2 ) for any q ∈ [p, +∞] and the embedding is independent of λ. Namely, there exists a constant C depending on q, p, and μ min such that for any (u, v) ∈ X λ , Proof. The proof is inspired by Lemma 2.5 in [20]. Suppose (u, v) ∈ X λ . For any vertex x 0 ∈ V , we have (12) which implies that X λ → L ∞ (V, R 2 ) continuously and the embedding is independent of λ. Consequently, we have X λ → L q (V ) continuously for any p ≤ q ≤ ∞. In fact, for any (u, v) ∈ X λ , we have (u, v) ∈ L p (V, R 2 ). Then, for any p ≤ q, Noting that X λ is reflexive, for a bounded sequence {(u k , v k )} in X λ , there exists (u, v) ∈ X λ and a subsequence that we still call {(u k , v k )} such that Vol. 25 (2023) Existence and convergence of solutions for p-Laplacian systemsPage 9 of 21 50 Then we get that, for any (ϕ, ψ) ∈ L p * (V, Take any x 0 ∈ V and let Obviously, they both belong to L p (V, R 2 ). By substituting (15) and (16) into (14), we get On the other hand, since This together with (18) gives that V (|u k | p + |v k | p )dμ ≤ ε when k is large enough. Therefore, For any x ∈ V , we have that Finally, for any p < q < ∞, we have that Namely, there exists a constant C depending only on Ω a , Ω b and q 1 , q 2 such that for Moreover, for any bounded sequence Proof. The proof is similar to Lemma 2.6 in [20] and we omit it here.

Existence of the ground state solutions
In this section, we prove the existence of ground state solutions of system (1) by the method of Nehari manifold. First, we state some important properties of N λ in the following lemma.  Proof. (i) For any (u, v) ∈ X λ \{(0, 0)}, we define the function Since F is a positively homogeneous function of degree γ, γ > p and (u, v) ≡ (0, 0), there exists t 0 > 0 such that g(t 0 ) = 0, which implies that By Remark 1.2 (ii) and Lemma 2.3, we have This gives that Next, we prove the crucial conclusion related to the nonlinear term F .
Proof. First, we claim that The proof of (21) is similar to that of Lemma 8 in [14]. For the convenience of the reader, we sketch it here. Indeed, by the mean value theorem, we get Then by Remark 1.2, we obtain For any given ε > 0, applying the Young's inequality to the last inequality, there exists C ε such that Note that and let then we obtain Thus, g k (x) → 0 for any x ∈ V . By the Lebesgue dominated convergence theorem, we have And then Therefore, we conclude that (21) holds.
Proof of Theorem 1.1. The proof is divided into two steps.
(i) m λ can be achieved by some By weak lower semi-continuity of the norm for X λ and Lemma 3.2, we have Now, we prove that (u λ , v λ ) ∈ N λ . Since {(u k , v k )} is bounded in X λ , there exists a positive constant C such that, up to a subsequence, dμ, by similar arguments as in Lemma 3.1(i), we get that there exists some t ∈ (0, 1) such that t(u λ , v λ ) ∈ N λ . Then we have This together with (26) gives that m λ is achieved by (u λ , v λ ).
obtain that α(s) achieves its minimum at s = 0. Therefore, which implies that (27) holds. Combining (i) and (ii), we complete the proof of Theorem 1.1.
Remark 3.1. We can get the existence of the ground state solution to the Dilrichlet problem (4) using arguments similar to the above proof and Theorem 1.2 is proved. On the other hand, by proving the convergence of the ground state solution of the system (1), we can also get a ground state solution of the limit problem (4). One can see the next section for more details.

The asymptotic behavior of solutions
In this section, we prove that the ground state solution (u λ , v λ ) of the system (1) converges to a ground state solution of the Dilrichlet problem (4) as λ → ∞, which also implies Theorem 1.2.
and either c ≥ C 1 or c = 0.
On the other hand, this together with (29) implies (28).
For any (u, v) ∈ X λ , by Lemma 2.3 Note that γ > p and take ε = Thus, (u k , v k ) λ < ε for k large enough. Then we have which implies (u k , v k ) λ → 0 and c = 0. The desired results are proved for Finally, we prove Theorem 1.3.
Proof of Theorem 1.3. We need to prove that for any sequence λ k → ∞, the corresponding (u λ k , v λ k ) ∈ N λ k satisfying J λ k (u λ k , v λ k ) = m λ k converges in X λ to a ground state solution (u, v) of (4) along a subsequence.

Concluding remarks
For a single equation, our recent works [9,10] have proved the existence and convergence of solutions using the variational method. However, the situation becomes more complicated for nonlinear systems, in which case it is difficult to verify that the energy functional satisfies the (P S) condition. This paper is an initiation of the p-Laplacian system on graphs. To inspire interested readers, we give several remarks and research prospects in the following.