On a class of nonlinear Schr\"odinger equation on finite graphs

TL;DR

This paper establishes existence results for a nonlinear Schrödinger equation on finite graphs by proving key inequalities and deriving conditions on parameters, thus extending and improving previous theoretical findings.

Contribution

It introduces a Trudinger-Moser inequality on graphs and derives new existence conditions for solutions under curvature-dimension constraints.

Findings

Proved Trudinger-Moser inequality on finite graphs.

Established an integral inequality under curvature-dimension conditions.

Identified parameter bounds for positive solutions to the nonlinear Schrödinger equation.

Abstract

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\Delta$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$ \left \{ \begin{array}{lcr} -\Delta u-\alpha u=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right. $$ on graph $G$, where $f(x,u):V\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear function and $\alpha$ is a parameter. Firstly, we prove the Trudinger-Moser inequality on graph $G$, and under the assumption that $G$ satisfies the curvature-dimension type inequality $CD(m, \xi)$, we prove an integral inequality on $G$. Then by using the two inequalities, we prove that there exists a positive solution to the nonlinear Schr$\ddot{o}$dinger type equation if $\alpha<\frac{2\lambda^{2}}{m(\lambda-\xi)}$, where $\lambda$ is the eigenvalue of the graph Laplacian. Our work provides remarkable improvements to the previous results.

Equations

$$\left\{\begin{array}[]{lcr}-\Delta u-\alpha u=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array}\right.$$

$$-\Delta u+b(x)u=f(x,u),u\in W^{1,2}(\Omega )$$

$$\displaystyle\left\{\begin{array}[]{lcr}-\Delta u-\alpha u=|u|^{p-2}u\ \ in\ \ \Omega^{\circ}\\ u=0\ \ on\ \ \partial\Omega\\ \end{array}\right.$$

$$\alpha <{\lambda }_1(\Omega ),$$

$${\lambda }_1(\Omega )=\underset{u\not\equiv 0,u\mid \partial \Omega =0}{\inf }\frac{{\int }_{\Omega }|\nabla u{|}^{2}d\mu }{{\int }_{\Omega }{u}^{2}d\mu }.$$

$$\displaystyle\left\{\begin{array}[]{lcr}-\Delta u-\alpha u=f(x,u),\\ u\in W^{1,2}(V)\\ \end{array}\right.$$

$$\Delta u(x)=\frac{1}{\mu (x)}\sum _{y\sim x}{\omega }_{xy}[u(y)-u(x)].$$

$$\Gamma (u,v)(x)=\frac{1}{2}\{\Delta (u(x)v(x))-u(x)\Delta v(x)-v(x)\Delta u(x)\},$$

$$\Gamma (u,v)(x)=\frac{1}{2\mu (x)}\sum _{y\sim x}{\omega }_{xy}(u(y)-u(x))(v(y)-v(x)).$$

$$|\nabla u|(x)=\sqrt{2\Gamma (u,u)(x)}=(\frac{1}{\mu (x)}\sum _{y\sim x}{\omega }_{xy}(u(y)-u(x){)}^2{)}^{1/2}.$$

$${\Gamma }_2(u,v)(x)=$$

$$=$$

$$-\frac{1}{2}\frac{1}{d_x}\sum _{y\sim x}{\mu }_{xy}[f(x)-f(y){]}^2+\frac{1}{2}[\frac{1}{d_x}\sum _{y\sim x}{\mu }_{xy}(f(x)-f(y)){]}^2.$$

$${\int }_{V}ud\mu =\sum _{x\in V}\mu (x)u(x).$$

$$\lambda ={\lambda }_G=\inf \frac{{\sum }_{x,y\in V,y\sim x}(u(y)-u(x){)}^2{\omega }_{xy}}{{\sum }_{x\in V}{u}^2(x)\mu (x)}=\inf \frac{{\int }_{\Omega }|\nabla u{|}^{2}d\mu }{{\int }_{\Omega }{u}^{2}d\mu }.$$

$$-\Delta u(x)=\frac{1}{\mu (x)}\sum _{y\sim x}{\omega }_{xy}[u(x)-u(y)]=\lambda u(x).$$

$${\Gamma }_2(u,u)(x)\ge \frac{1}{m}(\Delta u(x){)}^2+\xi \Gamma (u,u)(x).$$

$$exp(\beta |u{|}^{\frac{p}{p-1}})\in L^1(\Omega ).$$

$$\underset{||u|{|}_{W_0^{1,p}(\Omega )}\le 1}{\sup }{\int }_{\Omega }exp(\beta |u{|}^{\frac{p}{p-1}})dx\le C(p)|\Omega |if\beta \le {\beta }_p,$$

$$\underset{t\to +\infty }{\lim }\frac{f(x,t)}{exp(\beta |t{|}^{\frac{p}{p-1}})}=0.$$

$$\underset{||u|{|}_{W^{1,p}(V)}\le 1}{\sup }{\int }_{V}exp(\beta |u{|}^{\frac{p}{p-1}})d\mu \le C|V|forany\beta >1andp>2,$$

$$\frac{2{\lambda }^2}{m(\lambda -\xi )}{\int }_V{u}^{2}d\mu \le {\int }_V|\nabla u{|}^{2}d\mu .$$

$$||u|{|}_{\alpha }=({\int }_V(|\nabla u{|}^2-\alpha |u{|}^2)d\mu {)}^{\frac{1}{2}}.$$

$$||u|{|}_{W^{1,2}}=({\int }_V|\nabla u{|}^2{)}^{1/2}.$$

$$\alpha <\frac{2{\lambda }^2}{m(\lambda -\xi )},$$

$$||u|{|}_{W^{1,p}}=({\int }_V|\nabla u{|}^p{)}^{1/p}.$$

$$({\int }_V|u{|}^{q}d\mu {)}^{1/q}\le C({\int }_V|\nabla u{|}^{s}d\mu {)}^{1/s}.$$

$$({\int }_V|u{|}^{\frac{p}{p-1}}d\mu {)}^{\frac{p-1}{p}}\le C_0({\int }_V|\nabla u{|}^{p}d\mu {)}^{\frac{1}{p}}=C_0||u|{|}_{W^{1,p}(V)}\le C_0.$$

$$||u|{|}_{L^{\infty }(V)}\le \frac{C_0}{{\mu }_{min}}.$$

$$({\int }_{V}exp(\beta |u{|}^{\frac{p}{p-1}})d\mu {)}^{\frac{p-1}{p}}\le exp(\frac{\beta C_0}{{\mu }_{min}})|V{|}^{\frac{p-1}{p}}.$$

$$\underset{||u|{|}_{W^{1,p}(V)}\le 1}{\sup }{\int }_{V}exp(\beta |u{|}^{\frac{p}{p-1}})d\mu \le C|V|,$$

Full text

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on graph $G$, where $f(x,u):V\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear function and $\alpha$ is a parameter. Firstly, we prove the Trudinger-Moser inequality on graph $G$, and under the assumption that $G$ satisfies the curvature-dimension type inequality $CD(m,\xi)$, we prove an integral inequality on $G$. Then by using the two inequalities, we prove that there exists a positive solution to the nonlinear Schr$\ddot{o}$dinger type equation if $\alpha<\frac{2\lambda^{2}}{m(\lambda-\xi)}$, where $\lambda$ is the eigenvalue of the graph Laplacian. Our work provides remarkable improvements to the previous results.

The author is supported by the National Natural Science Foundation of China (Grant No. 11601368)

1. Introduction

During the past several decades, the nonlinear Schr$\ddot{o}$dinger type equation of the form

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has been extensively studied. In equation (1.1), $\Omega\sqsubseteq R^{n},n\geq 2,f(x,u):\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear continuous function and $b(x)\in C(\Omega,\mathbb{R})$ is a given potential. This type equation provides a good model for developing new mathematical methods and has important applications in science and engineering. Many papers are devoted to this kind of equations such as [14, 15, 16, 17].

Most recently, the investigation of discrete weighted Laplacians and various equations on graphs have attracted much attention. In [6], A. Grigor yan, Y. Lin and Y. Y. Yang proved that there exists a positive solution to

[TABLE]

on graphs for any $p>2$ if

[TABLE]

where

[TABLE]

For more results about differential equations on graphs, we can refer to [5, 7, 8, 9, 18, 19, 20].

In this paper, we consider a class of nonlinear Schr$\ddot{o}$dinger equation of the form

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on finite graph $G$. Here $f(x,u):V\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear function and $\alpha$ is a parameter. The equation (1.9) can be viewed as a discrete version of (1.1).

Firstly, we give some notations and settings. Let $G=(V,E)$ be a graph where $V$ denotes the vertex set and $E$ denotes the edge set. The degree of vertex $x$, denoted by $\mu(x)$, is the number of edges connected to $x$. If $V$ contains finite vertexes, we say that $G$ is a finite graph. If for every vertex $x$ of $V$, $\mu(x)$ is finite, we say that $G$ is a locally finite graph. When a graph is finite, this graph is certainly locally finite. We denote $x\sim y$ if vertex $x$ is adjacent to vertex $y$. We use $(x,y)$ to denote an edge in $E$ connecting vertices $x$ and $y$. Let $\omega_{xy}=\omega_{yx}>0$ where $\omega_{xy}$ is the edge weight. The finite measure $\mu(x)=\sum_{y\sim x}\omega_{xy}$. A graph $G$ is called connected if for any vertices $x,y\in V$, there exists a sequence $\{x_{i}\}_{i=0}^{n}$ that satisfies $x=x_{0}\sim x_{1}\sim x_{2}\sim\cdot\cdot\cdot\sim x_{n}=y$.

From [10], for any function $u:V\rightarrow\mathbb{R}$, the $\mu$-Laplacian of $u$ is defined as

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The associated gradient form reads

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By (1.10) and (1.11), we also have

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The length of the gradient for $u$ is

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The Ricci curvature operator on graphs $\Gamma_{2}$ by iterating $\Gamma$ as

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To compare with the Euclidean setting, we denote, for any function $u:V\rightarrow\mathbb{R}$,

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From [3], the eigenvalue of the Laplacian $\Delta$ on $G=(V,E)$ can be defined as

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From (1.17) we can see all eigenvalues are nonnegative and we can get $0=\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}\leq\cdot\cdot\cdot\leq\lambda_{n-1}$. For more details, we can refer to [3]. By (1.16) and Lemma 1.10 in [3], let $u$ denote a harmonic eigenfunction achieving $\lambda$ in (1.17), then, for any vertex $x\in V$, we have

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In [10], Y. Lin and S. T. Yau introduced the curvature-dimension type inequality $CD(m,\xi)$ as follows:

Definition 1.1**.**

(Curvature-dimension type inequality) The operator $\Delta$ satisfies the curvature-dimension type inequality $CD(m,\xi)$ for some $m>1$ if for every $u$,

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We call $m$ the dimension of the operator $\Delta$ and $\xi$ the lower bound of the Ricci curvature of the operator $\Delta$.

It is easy to see that for $m<m^{\prime}$, the operator $\Delta$ satisfies the curvature-dimension type inequality $CD(m^{\prime},\xi)$ if it satisfies the curvature-dimension type inequality $CD(m,\xi)$.

In [10], Y. Lin and S. T. Yau proved that any locally finite graph satisfies either $CD(2,\frac{2}{d}-1)$ if $d$ is finite, or $CD(2,-1)$ if $d$ is infinite, where $d=\sup_{x\in V}\sup_{y\sim x}\frac{\mu(x)}{\mu_{xy}}$.

In addition to the above curvature-dimension type inequality, we will introduce the well-know Trudinger-Moser inequality. In [11], when $p>2$, it asserts that

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Moreover, there exists a constant $C=C(p)$ which depends only on $p$ such that

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where $p>2$ and $\beta_{p}=p(\omega_{p-1})^{\frac{1}{p-1}},\ \ and\ \ \omega_{p-1}$ is the measure of the unit sphere in $R^{p}$. In the third section, we will prove the Trudinger-Moser inequality in Theorem 1.3 on graphs as a discrete version of (1.21).

Motivated by the Trudinger-Moser inequalities, we have

Definition 1.2**.**

([12]) Suppose that $f(x,t):V\times\mathbb{R}\rightarrow\mathbb{R}$. We call that the function $f$ has subcritical growth at $+\infty$ , if for all $\beta>0$ and $p>2$,

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In this paper, suppose that $G=(V,E)$ is a connected finite graph that satisfies the curvature-dimension type inequality $CD(m,\xi)$. Firstly, we prove the Trudinger-Moser inequality and an integral inequality (in Theorem 1.4) on graph $G$. Then by using the two inequalities, we prove that there exists a positive solution to the nonlinear Schr$\ddot{o}$dinger type equation (1.9) if $\alpha<\frac{2\lambda^{2}}{m(\lambda-\xi)}$, extending the result of (1.5) in equation (1.4).

Now, we state our main theorems.

Theorem 1.3**.**

(Trudinger-Moser inequality on finite graphs) Suppose that $G=(V,E)$ is a finite graph. Then there exists a constant $C$ which depends only on $p$ and $V$ such that

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where $|V|=\int_{V}d\mu(x)=$ Vol $V$, and Vol $V$ denotes the volume of the graph $G$.

Theorem 1.4**.**

Suppose that $G=(V,E)$ is a finite graph that satisfies the curvature-dimension type inequality $CD(m,\xi)$, and $u$ is a harmonic eigenfunction of $-\Delta$ with eigenvalue $\lambda$. Assume $\lambda\neq 0$. Then the following inequality holds

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Remark 1.5**.**

We define the norm

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By using Theorem 1.4, as $\alpha<\frac{2\lambda^{2}}{m(\lambda-\xi)}$, $||u||_{\alpha}$ is equivalent to the norm

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Theorem 1.6**.**

*Suppose that $G=(V,E)$ is a finite graph that satisfies the curvature-dimension type inequality $CD(m,\xi)$. Assume that $f:V\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following hypotheses:

$(H1)$ For any $x\in V$, $f(x,t)$ is continuous in $t\in\mathbb{R}$;

$(H2)$ For all $(x,t)\in V\times[0,+\infty),f(x,t)\geq 0$, and $f(x,0)=0$ for all $x\in V$;

$(H3)$ $f(x,t)$ has subcritical growth at $+\infty$ , i.e. $f$ satisfies (1.22);

$(H4)$For any $x\in V$ and $p>2$, there holds $\lim_{t\rightarrow 0+}\frac{f(x,t)}{t^{p-1}}=0$;

$(H5)$ There exists $q>p>2$ and $s_{0}>0$ such that if $s\geq s_{0}$, then there holds $0<qF(x,s)<f(x,s)s$ for any $x\in V$, where $F(x,s)=\int_{0}^{s}f(x,t)dt$.

Then, for any $p>2$ and*

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there exists a positive solution to the equation (1.9).

Remark 1.7**.**

We consider $\lambda=\lambda_{1}$ in (1.27), where $\lambda_{1}$ is the first nonzero eigenvalue of $-\Delta$ of graph $G$. By Lemma 2.1 in the second section in this paper, we have that $\lambda_{1}\geq\frac{m\xi}{m-1}$. We can easily check that when $\frac{m\xi}{m-1}\leq\lambda_{1}<\frac{m\xi}{m-2}$, we have $\frac{2\lambda_{1}^{2}}{m(\lambda_{1}-\xi)}>\lambda_{1}$. So we provide a remarkable improvement to the result of (1.5) in equation (1.4). For example, consider a connected path with two vertices a and b. It has a nonzero eigenvalue $\lambda_{1}=2$, and satisfies CD(2,1). We can check that $\frac{2\lambda_{1}^{2}}{m(\lambda_{1}-\xi)}=4>\lambda_{1}=\frac{m\xi}{m-1}=2$.

This paper is organized as follows. In Section 2, we introduce some notations and lemmas which are useful for the proof of our main theorems. In section 3, section 4, and section 5, we prove our main theorems.

2. Preliminaries

In this section, we introduce some notations and lemmas. Throughout this paper, we denote $L^{p}(V)$ the Banach space with the norm $||u||_{L^{p}}=(\int_{V}|u|^{p})^{1/p}$, and $W^{1,p}(V)$ can be defined as a set of all functions $u:V\rightarrow\mathbb{R}$ under the norm

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From [2] (See Theorem 2.1), we have the following Lemma 2.1.

Lemma 2.1**.**

([2]) Suppose that $G=(V,E)$ is a finite graph that satisfies the curvature-dimension type inequality $CD(m,\xi)$, and the Ricci curvature of $G$ is at least $\xi$. Then any nonzero eigenvalue $\lambda$ of $-\Delta$ satisfies $\lambda\geq\frac{m\xi}{m-1}$.

Lemma 2.2**.**

(Ambrosetti-Rabinowitz [1]). Let $(X,||\cdot||)$ be a Banach space, $J\in C^{1}(X,\mathbb{R})$, $e\in X$ and $r>0$ such that $||e||>r$ and $b=\inf_{||u||=r}J(u)>J(0)>J(e)$. If J satisfies the $(PS)_{c}$ condition with $c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))$, where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=0,\gamma(1)=e\}$, then c is a critical value of J.

From [6](See Theorem 7 and Theorem 8), we have the following lemma.

Lemma 2.3**.**

Let $G=(V,E)$ be a finite graph. Then for all $s>1$, $W^{1,s}(V)$ is embedded in $L^{q}(V)$ for all $1\leq q\leq+\infty$. In particular, there exists a constant C depending only on $s$ and $V$ such that

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Moreover, $W^{1,s}(V)$ is pre-compact, namely, if $\{u_{k}\}$ is bounded in $W^{1,s}(V)$, then us to a subsequence, there exists some $u\in W^{1,s}(V)$ such that $u_{k}\rightarrow u$ in $W^{1,s}(V)$.

3. The proof of Theorem 1.3

Proof.

Let function $u$ satisfy $||u||_{W^{1,p}(V)}\leq 1$. Since $p>2$ and $\frac{p}{p-1}>1$, by Lemma 2.3, we obtain that there exist a constant $C_{0}$ such that

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Denote $\mu_{min}=min_{x\in V}\mu(x)$. Then (3.1) leads to

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Thus for any $\beta>1$ and $p>2$, we have

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So, we have

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where $C=(exp(\frac{\beta C_{0}}{\mu_{min}}))^{\frac{p}{p-1}}$.

∎

4. The proof of Theorem 1.4

Proof.

By (1.11), (1.12), (1.13) and (1.14), we have

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On the other hand, by (1.10), we have

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By (4.1) and (4.2), we have

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Supposing $u$ is a harmonic eigenfunction that satisfies $-\Delta u(x)=\lambda u(x)$, by (4.3), we have

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We consider

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On the other hand, since $G$ satisfies the curvature-dimension type inequality $CD(m,\xi)$, that is,

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we have

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By (4.5) and (4.7), we have

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By (4.8) and Lemma 2.1, since $\lambda>\xi$, we have

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∎

5. The proof of Theorem 1.6

Proof.

Let $p>2$ and $\alpha<\frac{2\lambda^{2}}{m(\lambda-\xi)}$ be fixed. Now, we define the functional

$J:W^{1,2}(V)\rightarrow R$ by

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where $u^{+}(x)=\max\{u(x),0\}$. Indeed, from (H4), there exist $\tau,\delta>0$ such that if $|u|\leq\delta$ we have

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On the other hand, by (H3), there exist $c,\beta$ such that

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Then we obtain that, for $q>p$ ,

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Combining 5.2 and 5.4, we obtain that

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By the H$\ddot{o}$lder inequality, we have

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where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. By the Trudinger-Moser inequality in Theorem 1.3, we obtain that

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By Lemma 2.3, there exists some constant C that depends only on p and $V$ such that

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Since $q>p>2$, by (5.6),(5.7) and (5.8), we can find some sufficiently small $r>0$ such that if $||u||_{W^{1,2}}=r$ we have

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Therefore

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By (H5), there exist two positive constants $c_{1}$ and $c_{2}$ such that

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Take $u_{0}\in W^{1,2}(V)$ such that $u_{0}\geq 0$ and $u_{0}\neq 0$. For any $t>0$, we have

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Since $q>p>2$, we have $J(tu_{0})\rightarrow-\infty$ as $t\rightarrow+\infty$. Hence there exists some $u_{1}\in W^{1,2}(V)$ satisfying

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Now we prove that $J(u)$ satisfies the $(PS)_{c}$ condition for any $c\in R$. To see this, we assume $J(u_{k})\rightarrow c$ and $J^{\prime}(u_{k})\rightarrow 0$ as $k\rightarrow\infty$, that is

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view of (H5), we obtain from (5.14) and (5.15) that $u_{k}$ is bounded in $W^{1,2}(V)$. Then the $(PS)_{c}$ condition follows by Lemma 2.3 . Combining (5.10), (5.13) and the obvious fact that $J(0)=0$, we conclude by Lemma 2.2 that there exists a function $u\in W^{1,2}(V)$ such that $J(u)=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J(\gamma(t))>0$ and $J^{\prime}(u)=0$, where $\Gamma=\{\gamma\in C([0,1],W^{1,2}(V)):\gamma(0)=0,\gamma(1)=u_{1}\}$. Hence there exists a nontrivial solution $u\in W^{1,2}(V)$ to the equation

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Testing (5.18) by $u^{-}=\min\{u,0\}$ and noting that

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we have

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This implies that $u^{-}\equiv 0$ and thus $u\geq 0$. ∎

Acknowledgments The author thanks the referees for their comments and time.