Abstract
Let be a connected finite graph and be the -Laplacian on with . We consider a perturbed -th Yamabe equation , where are functions with ; and are two positive constants. Using the variational method, we prove that there exists some positive constant such that for all , the above equation has two distinct solutions.
1. Introduction and Main Results
Let be a finite graph, where denotes the vertex set and denotes the edge set. Let be a finite measure and be the weight of an edge. The graph satisfies the following properties. (a)For any edge , and (symmetric)(b)For any there are only finite such that (locally finite)(c)For any , there exist finite edges connecting and (connected)(d)There exists a constant such that for all (uniformly positive measure)(e)The distance of two vertices is defined by the minimal number of edges which connect these two vertices. For a subset of , the distance is uniformly bounded from above for any (bounded domain)
To do various analysis works, some reasonable assumptions are made about the graph, which results in different prominent features of the graph in different contexts. For example, some similarities and differences in feature between the metric graph and the graph mentioned above can be found. The reader may refer to [1–3] and the references therein for more details.
For any function , , the -Laplacian of is defined as where denotes . is a nonlinear operator when .
In the case of Grigor’yan et al. used the mountain-pass theorem to establish the existence results for the Yamabe equation [4] and the Schrödinger equation [5] on graphs. They also used a direct method of variation and the method of upper and lower solutions to study the existence of solutions for the Kazdan-Warner equation [6] on graphs. Later, Keller and Schwarz [7] studied the Kazdan-Warner equation on canonically compact graphs. Zhang and Zhao [8] studied the convergence of ground state solutions for a nonlinear Schrödinger equation on graphs. In the case of Ge [9] studied the existence of solutions for the -th Yamabe equation on graphs. One may refer to [10–16] for more related works.
In this paper, we consider the multiplicity of solutions to a -th Yamabe equation on a graph. For any function , the integral of over with respect to the vertex weight is defined by
Set
For any function defined on the edge set , the integral of over with respect to the edge weight is defined by
For any function , where is defined on the edge set , and for each edge .
Define
The well-known Yamabe equation [17, 18] derives from the Yamabe problem: Given a compact Riemannian manifold of dimension , find a metric conformal to with constant scalar curvature, which is to prove that there is a real number and a function satisfying the above Yamabe equation, where are functions on with and
In [9], Ge studied the following -th discrete Yamabe equation on a finite graph , where are functions with Using a direct method of variation, the author showed that (8) always has a positive solution for some constant We consider the following -th Yamabe equation where are functions with ; and are positive constants. Note that, in equation (9), we add a perturbed term In order to make our derivation possible, we have to set . By using the mountain-pass theorem, which is due to Ambrosetti and Rabinowitz [19], and a direct method of variation, we prove that (9) has two distinct solutions. Now, we can state the theorem as follows.
Theorem 1. Let be a finite graph and be functions with . Assume that . Then, there exists such that for any , (9) has two distinct solutions.
In case , we have the following result.
Corollary 2. Let be a finite graph and be functions with . Assume that . Then, there exists such that for any , the following Yamabe equation has two distinct solutions.
The multiplicity of solutions to certain equations on a graph was extensively studied by Grigor’yan et al. [5], Liu and Yang [12], Huang et al. [20], and Liu [21]. More results have been obtained in the Euclidean space; we refer the reader to [22–27] and the references therein.
2. Preliminaries
Define a Sobolev space and a norm on it by
Since is a finite graph, then is exactly the set of all functions on , a finite dimensional linear space. This implies the following Sobolev embedding.
Lemma 3 (Sobolev embedding theorem, see [4]). Let be a finite graph and . Then, is embedded in for all In particular, there exists a constant depending only on and such that
for all and for all Moreover, the Sobolev space is precompact; namely, if is bounded in , then there exists some such that, up to a subsequence, in .
The functional related to (9) is
The existence of solutions of (9) is transformed into finding the critical points of .
Let be a Banach space; we say that satisfies the condition for some real number if, for any sequence of functions such that and for all as there holds up to a subsequence in . To prove Theorem 1, we need the following mountain-pass theorem.
Theorem 4 (mountain-pass theorem, see [19]). Let be a Banach space, , and be such that and If satisfies the condition with where then is a critical value of
3. Proof of the Main Results
Lemma 5. There exist positive constants and such that for all with if for a sufficiently small
Proof. Let . For , by the Hölder inequality, we have
where is a constant depending on .
Let . By Lemma 3, there exists some constant depending on and such that
From (16) and (17), and noting that , we have
where . Let , then . For , we have
Thus, there exists some sufficiently small such that and
Let ; we get for
Lemma 6. satisfies the condition for any real number
Proof. For any , take such that and for all as . Namely,
for all
Taking as the test function in (22), we have
From (21) and (23), we obtain that
which implies that
Suppose that is unbounded in . For we have
as , which contradicts (25). Hence, is bounded in .
Taking a function with and passing to the limit , we have
It is obvious that , with ; for some with Moreover, satisfies the condition with where
and is a critical value of Thus, there exists a solution in such that
Next, we prove that there exists another solution such that , where is another critical value of .
Lemma 7. There exist some and with such that if
Proof. Consider the equation in . Define the functional Note that where is a sufficiently small constant; is a constant depending on ; and we use Young’s inequality in the proof of the first inequality. Hence, has a lower bound in for a sufficiently small . Let and taking a sequence satisfies as Moreover, is bounded in . By Lemma 3, there exists some up to a subsequence in . Then, and is a solution of (29). It follows that Now, we consider the derivative of By (33), we get Let , and we finish the proof.
Now, we prove that there exists another solution with such that for , where By Lemma 5, we know that has a lower bound on . By Lemma 7, we get that
Take the sequence with such that as Since is bounded in , by Lemma 3, there exists some up to a subsequence in . Moreover,
Then, and is the minimizer of on . Lemma 5 implies that Calculating the Euler–Lagrange equation of for , we get that
Hence,
Thus, is a solution of (9). This ends the proof of Theorem 1.
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We would like to thank Professor Desheng Li for many helpful conversations.