# Multiple Solutions of ap-th Yamabe Equation on Graph

Authors: Zhongqi Peng; Aimin Zhu; Tingting Zhang
Journal: Journal of Function Spaces
 (2023)

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## Abstract

LetG=V,E be a connected finite graph and Δp be the p-Laplacian on G with p>1. We consider a perturbed p-th Yamabe equation −Δpu−λup−2u=huα−2u+εf, where h,f:V⟶ℝ are functions with h,f>0; 1<p<α;λ and ε are two positive constants. Using the variational method, we prove that there exists some positive constant ϵ1 such that for all ϵ∈0,ϵ1, the above equation has two distinct solutions.


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## Body

## 1. Introduction and Main Results

LetG=V,E be a finite graph, where V denotes the vertex set and E denotes the edge set. Let μ:V⟶ℝ+ be a finite measure and w:E⟶ℝ+ be the weight of an edge. The graph G satisfies the following properties.
(a)
For any edgeij∈E, wij>0 and wij=wji (symmetric)(b)
For anyi∈V, there are only finite j∈V such that ij∈E (locally finite)(c)
For anyi,j∈V, there exist finite edges connecting i and j (connected)(d)
There exists a constantμmin>0 such that μi≥μmin for all i∈V (uniformly positive measure)(e)
The distancedij of two vertices i,j∈V is defined by the minimal number of edges which connect these two vertices. For a subset Ω of V, the distance dij is uniformly bounded from above for any i,j∈Ω (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 functionu:V⟶ℝ, p>1, the p-Laplacian of u is defined as
(1)Δpui=1μi∑j~iwijuj−uip−2uj−ui,where j~i denotes ij∈E. Δp is a nonlinear operator when p≠2.In the case ofp=2, 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 p>1, Ge [9] studied the existence of solutions for the p-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 ap-th Yamabe equation on a graph. For any function u:V⟶ℝ, the integral of u over V with respect to the vertex weight μ is defined by
(2)∫Vudμ=∑i∈Vμiui.Set(3)∫Vdμ=VolG.For any functiong defined on the edge set E, the integral of g over E with respect to the edge weight w is defined by
(4)∫Egdw=∑i~jwijgij.For any functionu:V⟶ℝ,
(5)∫E∇updw=∑i~jwijuj−uip,where ∇u is defined on the edge set E, and ∇uij=uj−ui for each edge i~j.Define(6)λ1=infu≢0∫E∇updw+∫Vupdμ2∫Vupdμ.The well-known Yamabe equation [17, 18]
(7)Δu+gu=khuN−1derives from the Yamabe problem: Given a compact Riemannian manifold M,l of dimension n≥3, find a metric conformal to l with constant scalar curvature, which is to prove that there is a real number k and a function u>0 satisfying the above Yamabe equation, where g,h are functions on M with h>0 and N=2n/n−2.In [9], Ge studied the following p-th discrete Yamabe equation
(8)Δpu+gup−1=khuα−1on a finite graph G, where g,h are functions with h>0;α≥p>1. Using a direct method of variation, the author showed that (8) always has a positive solution for some constant k. We consider the following p-th Yamabe equation
(9)−Δpu−λup−2u=huα−2u+ϵf,where h,f:V⟶ℝ are functions with h,f>0; 1<p<α;λ and ε are positive constants. Note that, in equation (9), we add a perturbed term ϵf. In order to make our derivation possible, we have to set g≡λ. 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.
LetG=V,E be a finite graph and h,f:V⟶ℝ be functions with h,f>0. Assume that 1<p<α,1<λ<λ1. Then, there exists ϵ1>0 such that for any ϵ∈0,ϵ1, (9) has two distinct solutions.In casep=2, we have the following result.Corollary 2.
LetG=V,E be a finite graph and h,f:V⟶ℝ be functions with h,f>0. Assume that α>2,1<λ<λ1. Then, there exists ϵ1>0 such that for any ϵ∈0,ϵ1, the following Yamabe equation
(10)−Δu−λu=huα−2u+ϵfhas 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(11)W1,pG=u:V⟶ℝ∫E∇updw+∫Vupdμ<+∞,uW1,pG=∫E∇updw+∫Vupdμ1/p.SinceG is a finite graph, then W1,pG is exactly the set of all functions on V, a finite dimensional linear space. This implies the following Sobolev embedding.Lemma 3 (Sobolev embedding theorem, see [4]).
LetG=V,E be a finite graph and p>1. Then, W1,pG is embedded in LqG for all 1≤q≤+∞. In particular, there exists a constant Cp,G depending only on p and G such that
(12)uLqG≤Cp,GuW1,pG,for all 1≤q≤+∞ and for all u∈W1,pG. Moreover, the Sobolev space W1,pG is precompact; namely, if un is bounded in W1,pG, then there exists some u∈W1,pG such that, up to a subsequence, un⟶u in W1,pG.
The functional related to (9) is
(13)Jϵu=1p∫E∇updw−λp∫Vupdμ−1α∫Vhuαdμ−ϵ∫Vfudμ.
The existence of solutions of (9) is transformed into finding the critical points of Jε.
LetX,· be a Banach space; we say that J:X⟶ℝ satisfies the PSc condition for some real number c if, for any sequence of functions un:X⟶ℝ such that Jun⟶c and J′un⟶0 for all ϕ∈W1,pG as n⟶+∞, there holds up to a subsequence un⟶u in X. To prove Theorem 1, we need the following mountain-pass theorem.Theorem 4 (mountain-pass theorem, see [19]).
LetX,· be a Banach space, J∈C1X,ℝ,e∈X, and r>0 be such that e>r and
(14)b≔infu=rJu>J0≥Je.
IfJ satisfies the PSc condition with c≔infγ∈Γmaxt∈0,1Jγt, where
(15)Γ≔γ∈C0,1,X:γ0=0,γ1=e,then c is a critical value of J.


## 3. Proof of the Main Results

Lemma 5.
There exist positive constantsrϵ and δϵ such that Jϵu≥δϵ for all u∈W1,pG with rϵ/p≤uW1,pG≤rϵ if 0<ϵ<ϵ1 for a sufficiently small ϵ1.Proof.
LetfM=maxi∈Vfi>0. For p>1, by the Hölder inequality, we have
(16)∫Vfudμ≤fM∫V1p/p−1dμp−1/p∫Vupdμ1/p≤fMVolGp−1/puW1,pG=Cf,p,GuW1,pG,where Cf,p,G>0 is a constant depending on f,p,G.
LethM=maxi∈Vhi>0. By Lemma 3, there exists some constant Cp,G depending on p and G such that
(17)∫Vhuαdμ≤hMuLαGα≤Cp,GαhMuW1,pGα.
From (16) and (17), and noting that 1<λ<λ1, we have
(18)Jϵu≥1p∫E∇updw−1pλλ1uW1,pGp−∫Vupdμ−Cp,GαhMαuW1,pGα−ϵCf,p,GuW1,pG≥uW1,pGτpuW1,pGp−1−Cp,GαhMαuW1,pGα−1−ϵCf,p,G,where τ=λ1−λ/λ1. Let rϵ=ϵ1/p, then 1/pϵ1/p≤uW1,pG≤ϵ1/p. For 1<p<α, we have
(19)limϵ⟶0+τ/ppϵp−1/p−Cp,GαhM/αϵα−1/p−ϵCf,p,Gτ/ppϵp−1/p=1.
Thus, there exists some sufficiently smallϵ1 such that 0<ϵ<ϵ1 and
(20)τppϵp−1/p−Cp,GαhMαϵα−1/p−ϵCf,p,G≥τ2pp+1ϵp−1/p.
Letδϵ=τϵ/2pp+2; we get Jϵu≥δϵ for 0<ϵ<ϵ1.Lemma 6.
Jϵ satisfies the PSc condition for any real number c.Proof.
For anyc∈ℝ, take un⊂W1,pG such that Jϵun⟶c and J′ϵunϕ⟶0 for all ϕ∈W1,pG as n⟶+∞. Namely,
(21)1p∫E∇unpdw−λp∫Vunpdμ−1α∫Vhunαdμ−ϵ∫Vfundμ=c+on1,(22)∫V−Δpun−λunp−2un−hunα−2un−ϵfϕdμ=on1ϕW1,pG,for all ϕ∈W1,pG.
Takingun as the test function ϕ in (22), we have
(23)∫E∇unpdw−λ∫Vunpdμ−∫Vhunαdμ−ϵ∫Vfundμ=on1unW1,pG.
From (21) and (23), we obtain that
(24)α−ppα∫E∇unpdw=λα−ppα∫Vunpdμ+ϵα−1α∫Vfundμ+c+on1unW1,pG+on1≤α−ppαλλ1unW1,pGp−∫Vunpdμ+ϵα−1αCf,p,GunW1,pG+c+on1,which implies that
(25)τα−ppαunW1,pGp≤ϵα−1αCf,p,GunW1,pG+c+on1.
Suppose thatun is unbounded in W1,pG. For 1<p<α, we have
(26)τα−ppαunW1,pGp−ϵα−1αCf,p,GunW1,pG−c+on1⟶+∞,as n⟶+∞, which contradicts (25). Hence, un is bounded in W1,pG.
Taking a functionu∗∈W1,pG with u∗≡0 and passing to the limit t⟶+∞, we have
(27)Jϵtu∗=tpp∫E∇u∗pdw−λtpp∫Vu∗pdμ−tαα∫Vhu∗αdμ−tϵ∫Vfu∗dμ⟶−∞.
It is obvious thatJϵ∈C1W1,pG,·, Jϵ0=0;Jϵu≥δϵ>0 with uW1,pG=rϵ/p; Jϵu~<0 for some u~ with u~W1,pG>rϵ/p. Moreover, Jϵ satisfies the PSc condition with c¯=minγ∈Γmaxt∈0,1Jϵγt, where
(28)Γ=γ∈C0,1,W1,pG:γ0=0,γ1=u~,and c¯ is a critical value of Jϵu. Thus, there exists a solution u¯ in W1,pG such that Jϵu¯=c¯≥δϵ>0.Next, we prove that there exists another solutionu^ such that Jϵu^=c^<0, where c^ is another critical value of Jϵu.Lemma 7.
There exist someρ and u∈W1,pG with uW1,pG=1 such that Jϵtu<0 if 0<t<ρ.Proof.
Consider the equation(29)−Δpu−λup−2u=f,in W1,pG. Define the functional
(30)Jfu=1p∫E∇updw−λp∫Vupdμ−∫Vfudμ.
Note that(31)Jfu≥1puW1,pGp−λpλ1uW1,pGp−ηp∫Vupdμ−Cp,q,η,f≥τ−ηpuW1,pGp−Cp,q,η,f,where 1/p+1/q=1;η>0 is a sufficiently small constant; Cp,q,η,f is a constant depending on p,q,η,f; and we use Young’s inequality in the proof of the first inequality. Hence, Jf has a lower bound in W1,pG for a sufficiently small η. Let mf=infu∈W1,pGJfu and taking a sequence un satisfies Jfun⟶mf as n⟶+∞. Moreover, un is bounded in W1,pG. By Lemma 3, there exists some u0∈W1,pG up to a subsequence un⟶u0 in W1,pG. Then,
(32)Jfu0=limn⟶+∞Jfun=mf,and u0 is a solution of (29). It follows that
(33)∫E∇u0pdw−λ∫Vu0pdμ=∫Vfu0dμ≥τu0W1,pG>0.
Now, we consider the derivative ofJϵtu0:(34)ddtJϵtu0=tp−1∫E∇u0pdw−λtp−1∫Vu0pdμ−tα−1∫Vhu0αdμ−ϵ∫Vfu0dμ.
By (33), we get
(35)ddtt=0Jϵtu0<0.
Letu=u0/u0W1,pG, and we finish the proof.Now, we prove that there exists another solutionu^∈W1,pG with u^W1,pG<rϵ/p such that
(36)Jϵu^=c^=infuW1,pG≤rϵJϵu<0,for 0<ϵ<ϵ1, where rϵ=ϵ1/p. By Lemma 5, we know that Jϵu has a lower bound on Brϵ=u∈W1,pG:uW1,pG≤rϵ. By Lemma 7, we get that infuW1,pG≤rϵJϵu=c^<0.Take the sequenceun⊂W1,pG with unW1,pG≤rϵ such that Jϵun⟶c^ as n⟶+∞. Since un is bounded in W1,pG, by Lemma 3, there exists some u^∈W1,pG up to a subsequence un⟶u^ in W1,pG. Moreover,
(37)limn⟶+∞unW1,pG=u^W1,pG,limn⟶+∞∫Vunpdμ=∫Vu∧pdμ,limn⟶+∞ϵ∫Vfundμ=ϵ∫Vfu^dμ,limn⟶+∞∫Vhunαdμ=∫Vhu∧αdμ.Then,(38)Jϵu^=limn⟶+∞Jϵun=c^<0,and u^ is the minimizer of Jϵu on Brϵ. Lemma 5 implies that u^W1,pG<rϵ/p. Calculating the Euler–Lagrange equation of Jϵu^ for ϕ∈W1,pG, we get that
(39)0=ddtt=0Jϵu^+tϕ=∫V−Δpu^−λu∧p−2u^−hu∧α−2u^−ϵfϕdμ.Hence,(40)−Δpu^−λu∧p−2u^=hu∧α−2u^+ϵf.Thus,u^ is a solution of (9). This ends the proof of Theorem 1.



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