# Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs – Boundary Value Problems

#### ByXuechen Zhang, Xingyong Zhang, Junping Xie and Xiaoli Yu

May 10, 2022

Note that the space (W:=W^{m_{1},p}(V)times W^{m_{2},q}(V)) with the norm (|(u,v)|=|u|_{W^{m_{1},p}(V)}+|v|_{W^{m_{2},q}(V)}) is a finite-dimensional Banach space. Consider the functional (varphi :Wto mathbb{R}) defined as

begin{aligned}[b] varphi (u,v)={}&frac{1}{p} int _{V}bigl( biglvert nabla ^{m_{1}}u bigrvert ^{p}+h_{1}(x) vert u vert ^{p}bigr),dmu +frac{1}{q} int _{V}bigl( biglvert nabla ^{m_{2}}v bigrvert ^{q}+h_{2}(x) vert v vert ^{q}bigr),dmu \ &{}- int _{V} F(x,u,v),dmu . end{aligned}

(3.1)

Then (varphi in C^{1}(W,mathbb{R})), and

begin{aligned} bigllangle varphi ‘(u,v),(phi _{1},phi _{2})bigrrangle =& int _{V} bigl[(text{pounds}_{m_{1},p}u,phi _{1})+bigl(h_{1}(x) vert u vert ^{p-2}u,phi _{1}bigr)-bigl( F_{u}(x,u,v), phi _{1}bigr) bigr],dmu \ &{}+ int _{V} bigl[(text{pounds}_{m_{2},q}v,phi _{2})+bigl(h_{2}(x) vert v vert ^{q-2}v, phi _{2}bigr)-bigl(F_{v}(x,u,v), phi _{2}bigr) bigr],dmu end{aligned}

(3.2)

for all ((u,v),(phi _{1},phi _{2})in W). Then ((u,v)in W) is a critical point of φ if and only if

$$int _{V} bigl(bigl(text{pounds}_{m_{1},p}u+h_{1}(x) vert u vert ^{p-2}u-F_{u}(x,u,v)bigr), phi _{1} bigr),dmu =0$$

and

$$int _{V} bigl(bigl(text{pounds}_{m_{2},q}v+h_{2}(x) vert v vert ^{q-2}v-F_{v}(x,u,v)bigr), phi _{2} bigr),dmu =0.$$

By the arbitrariness of (phi _{1}) and (phi _{2}) we conclude that

begin{aligned}& text{pounds}_{m_{1},p}u+h_{1}(x) vert u vert ^{p-2}u=F_{u}(x,u,v), \& text{pounds}_{m_{2},q}v+h_{2}(x) vert v vert ^{q-2}v=F_{v}(x,u,v). end{aligned}

Thus the problem of finding the solutions of system (1.1) is reduced to finding the critical points of the functional φ on W.

### Lemma 3.1

Assume that ((F_{4})) holds. Then the functional φ satisfies condition ((C)), that is, ({(u_{k},v_{k})}) has a convergent subsequence in W whenever (varphi (u_{k},v_{k})) is bounded and (|varphi ‘(u_{k},v_{k})|times (1+| (u _{k},v_{k})|)rightarrow 0) as (krightarrow infty ).

### Proof

Let ({(u_{k},v_{k})}) be a sequence in W such that (varphi (u_{k},v_{k})) is bounded and (|varphi ‘(u_{k},v_{k})|(1+|( u_{k},v_{k})|)rightarrow 0) as (krightarrow infty ). Then there exists a positive constant L such that

$$biglvert varphi (u_{k},v_{k}) bigrvert leq L, biglVert varphi ‘(u_{k},v_{k}) bigrVert bigl(1+ biglVert (u_{k},v_{k}) bigrVert bigr)leq L$$

for every (kin mathbb{N}). By ((F_{4})),there are constants (C_{1}>0) and (delta _{1}>0) such that

$$F_{t}(x,t,s)t+F_{s}(x,t,s)s-max {p,q}F(x,t,s)geq C_{1}bigl( vert t vert ^{ gamma _{1}}+ vert s vert ^{gamma _{2}}bigr)>0$$

for all (|(t,s)|>delta _{1}) and (xin V). Therefore

$$F_{t}(x,t,s)t+F_{s}(x,t,s)s-max {p,q} F(x,t,s) geq C_{1}bigl( vert t vert ^{ gamma _{1}}+ vert s vert ^{gamma _{2}}bigr)-C_{2}$$

for all ((t, s)in mathbb{R}^{2}) and (xin V), where

begin{aligned} C_{2} =& C_{1}max bigl{ vert t vert ^{gamma _{1}}+ vert s vert ^{gamma _{2}}mid biglvert (t,s) bigrvert le delta _{1} bigr} \ &{}+max bigl{ F_{t}(x,t,s)t+F_{s}(x,t,s)s- max {p,q} F(x,t,s)mid biglvert (t,s) bigrvert le delta _{1} bigr} . end{aligned}

Then for all large k, we have

begin{aligned}& bigl(max {p,q}+1bigr)L \& quad geq max {p,q}varphi (u_{k},v_{k})-bigl( varphi ‘(u_{k},v_{k}),(u_{k},v_{k}) bigr) \& quad = max {p,q} biggl[frac{1}{p} int _{V}bigl( biglvert nabla ^{m_{1}}u_{k} bigrvert ^{p}+h_{1}(x) vert u_{k} vert ^{p}bigr),dmu \& qquad {}+frac{1}{q} int _{V}bigl( biglvert nabla ^{m_{2}}v_{k} bigrvert ^{q}+h_{2}(x) vert v_{k} vert ^{q}bigr),dmu – int _{V} F(x,u_{k},v_{k}),dmu biggr] \& qquad {}- int _{V}( text{pounds}_{m_{1},p}u_{k},u_{k}),dmu – int _{V}h_{1}(x) vert u_{k} vert ^{p},dmu – int _{V}(text{pounds}_{m_{2},q}v_{k},v_{k}),dmu \& qquad {} – int _{V}h_{2}(x) vert v_{k} vert ^{p},dmu + int _{V}F_{u_{k}}(x,u_{k},v_{k})u_{k},dmu + int _{V}F_{v_{k}}(x,u_{k},v_{k})v_{k},dmu . end{aligned}

(3.3)

When (max {p,q}=p),

begin{aligned} (p+1)L geq & biggl(frac{p}{q}-1 biggr) int _{V}bigl( biglvert nabla ^{m_{2}}v_{k} bigrvert ^{q}+h_{2}(x) vert v_{k} vert ^{q}bigr),dmu \ & {}+ int _{V}bigl[bigl(F_{u_{k}}(x,u_{k},v_{k}),u_{k} bigr)+bigl(F_{v_{k}}(x,u_{k},v_{k}),v_{k} bigr)-pF(x,u_{k},v_{k})bigr],dmu \ geq & biggl(frac{p}{q}-1 biggr) int _{V}bigl( biglvert nabla ^{m_{2}}v_{k} bigrvert ^{q}+h_{2}(x) vert v_{k} vert ^{q}bigr),dmu \ &{}+ int _{V} C_{1}bigl( vert u_{k} vert ^{gamma _{1}}+ vert v_{k} vert ^{gamma _{2}}bigr),dmu -C_{2}sum _{xin V}mu (x) \ = & biggl(frac{p}{q}-1 biggr) Vert v_{k} Vert ^{q}_{W^{m_{2},q}(V)}+C_{1} int _{V}bigl( vert u_{k} vert ^{gamma _{1}}+ vert v_{k} vert ^{gamma _{2}}bigr),dmu -C_{2} sum_{xin V}mu (x). end{aligned}

Therefore (|v_{k}|_{W^{m_{2},q}(V)}), (|u_{k}|_{L^{gamma _{1}}(V)}), and (| v_{k}|_{L^{gamma _{2}}(V)}) are bounded. Since ((W,|cdot |)) is a finite-dimensional space, there exist positive constants (D_{1}) and (D_{2}) such that

$$Vert u_{k} Vert _{W^{m_{1},p}(V)}leq D_{1} Vert u_{k} Vert _{L^{gamma _{1}}(V)},qquad Vert v_{k} Vert _{W^{m_{2},q}(V)}leq D_{2} Vert v_{k} Vert _{L^{gamma _{2}}(V)}.$$

(3.4)

Thus (|u_{k}|_{W^{m_{1},p}(V)}) and (|v_{k}|_{W^{m_{2},q}(V)}) are bounded. So ({(u_{k},v_{k})}) is bounded in W. Similarly, when (max {p,q}=q), we can also prove that ({(u_{k},v_{k})}) is bounded in W. To sum up, ({(u_{k},v_{k})}) is bounded in W. Since W is of finite dimension, there is a convergent subsequence of ({(u_{k},v_{k})}). Hence φ satisfies the ((C))-condition. □

### Lemma 3.2

There exists a constant (rho >0) such that (varphi |_{partial B_{rho}(0)}> 0 ), where (B_{rho}={(u,v)in W:|(u,v)|_{W}<rho }).

### Proof

By ((F_{2})) there are (0< C_{4}<min {frac{1}{pK_{1}^{P}},frac{1}{qK_{2}^{q}} }) and a positive constant (delta _{2}< C_{3}), where (C_{3}=max {frac{1}{mu _{min} h_{1,min}}, frac{1}{mu _{min} h_{2,min}} }), such that

$$biglvert F(x,t,s) bigrvert leq C_{4} bigl( vert t vert ^{p}+ vert s vert ^{q} bigr)$$

(3.5)

for all (|(t,s)|leq delta _{2}). By Lemma 2.4 we have

$$Vert u Vert _{L^{p}(V)}leq K_{1} Vert u Vert _{W^{m_{1},p}(V)},qquad Vert v Vert _{L^{q}(V)}leq K_{2} Vert v Vert _{W^{m_{2},q}(V)},$$

(3.6)

where (K_{1}), (K_{2}) is defined in ((F_{2})). For every ((u,v)in W) with (|(u,v)|=rho =delta _{2} C_{3}^{-1}<1), by Lemma 2.3 we have

$$biglVert (u,v) bigrVert _{infty}le Vert u Vert _{infty}+ Vert v Vert _{infty}le C_{3} bigl( Vert u Vert _{W^{m_{1},p}(V)}+ Vert v Vert _{W^{m_{2},q}(V)} bigr)=delta _{2}.$$

Then by (3.5) and (3.6), for all ((u,v)in W) with (|(u,v)|=rho ), we have

begin{aligned}& varphi (u,v) \& quad = frac{1}{p} int _{V}bigl( biglvert nabla ^{m_{1}}u bigrvert ^{p}+h_{1}(x) vert u vert ^{p}bigr),dmu +frac{1}{q} int _{V}bigl( biglvert nabla ^{m_{2}}v bigrvert ^{q}+h_{2}(x) vert v vert ^{q}bigr),dmu – int _{V} F(x,u,v),dmu \& quad geq frac{1}{p} int _{V}bigl( biglvert nabla ^{m_{1}}u bigrvert ^{p}+h_{1}(x) vert u vert ^{p}bigr),dmu +frac{1}{q} int _{V}bigl( biglvert nabla ^{m_{2}}v bigrvert ^{q}+h_{2}(x) vert v vert ^{q}bigr),dmu \& qquad {} -C_{4} int _{V}bigl( vert u vert ^{p}+ vert v vert ^{q}bigr),dmu \& quad geq biggl(frac{1}{p}-K_{1}^{p}C_{4} biggr) int _{V}bigl( biglvert nabla ^{m_{1}}u bigrvert ^{p}+h_{1}(x) vert u vert ^{p}bigr),dmu \& qquad {}+ biggl(frac{1}{q}-K_{2}^{q}C_{4} biggr) int _{V}bigl( biglvert nabla ^{m_{2}}v bigrvert ^{q}+h_{2}(x) vert v vert ^{q}bigr),dmu \& quad = biggl(frac{1}{p}-K_{1}^{p}C_{4} biggr) Vert u Vert _{W^{m_{1},p}(V)}^{p} + biggl( frac{1}{q}-K_{2}^{q}C_{4} biggr) Vert v Vert _{W^{m_{2},q}(V)}^{q} \& quad geq min biggl{ biggl(frac{1}{p}-K_{1}^{p}C_{4} biggr) , biggl(frac{1}{q}-K_{2}^{q}C_{4} biggr) biggr} cdot textstylebegin{cases} frac{1}{2^{p-1}}( Vert u Vert _{W^{m_{1},p}(V)}+ Vert v Vert _{W^{m_{2},q}(V)})^{p} & text{if }pgeq q, \ frac{1}{2^{q-1}}( Vert u Vert _{W^{m_{1},p}(V)}+ Vert v Vert _{W^{m_{2},q}(V)})^{q} & text{if }p< q end{cases}displaystyle \& quad geq min biggl{ biggl(frac{1}{p}-K_{1}^{p}C_{4} biggr) , biggl(frac{1}{q}-K_{2}^{q}C_{4} biggr) biggr} cdot textstylebegin{cases} frac{rho ^{p}}{2^{p-1}}& text{if }pgeq q, \ frac{rho ^{q}}{2^{q-1}}& text{if }p< q end{cases}displaystyle \& quad := alpha >0. end{aligned}

The proof is completed. □

### Lemma 3.3

Assume that ((F_{1})) and ((F_{3})) hold. Then there exists ((u_{0},v_{0})in Wbackslash bar{B}_{rho} (0)) such that (varphi (u_{0},v_{0})leq 0 ).

### Proof

Choose (e=(e_{1},e_{2}) in W) such that (|e_{1}|_{L^{p}(V)}neq 0) and (|e_{2}|_{L^{q}(V)}neq 0). By ((F_{3})) there exist (varepsilon _{1}>0) and (delta _{3}>0) such that

$$F(x,t,s)geq biggl(frac{1}{p} frac{ Vert e_{1} Vert ^{p}_{W^{m_{1},p}(V)}}{ Vert e_{1} Vert ^{p}_{L^{p}(V)}}+ frac{1}{q} frac{ Vert e_{2} Vert ^{q}_{W^{m_{2},q}(V)}}{ Vert e_{2} Vert ^{q}_{L^{q}(V)}}+ frac{varepsilon _{1}}{2} biggr) bigl( vert t vert ^{p}+ vert s vert ^{q} bigr)$$

for all (|(t,s)|>delta _{3}) and (xin V). Thus by ((F_{1})) there exists (C_{5}>0) such that for all ((t, s)in mathbb{R}^{2}) and all (xin V),

$$F(x,t,s)geq biggl(frac{1}{p} frac{ Vert e_{1} Vert ^{p}_{W^{m_{1},p}(V)}}{ Vert e_{1} Vert ^{p}_{L^{p}(V)}}+ frac{1}{q} frac{ Vert e_{2} Vert ^{q}_{W^{m_{2},q}(V)}}{ Vert e_{2} Vert ^{q}_{L^{q}(V)}}+ frac{varepsilon _{1}}{2} biggr) bigl( vert t vert ^{p}+ vert s vert ^{q} bigr)-C_{5}.$$

Then for every (lambda >0), we have

begin{aligned} varphi (lambda e_{1},lambda e_{2}) =& frac{1}{p} int _{V}bigl( biglvert nabla ^{m_{1}} lambda e_{1} bigrvert ^{p}+h_{1}(x) vert lambda e_{1} vert ^{p}bigr),dmu + frac{1}{q} int _{V}bigl( biglvert nabla ^{m_{2}}lambda e_{2} bigrvert ^{q}+h_{2}(x) vert lambda e_{2} vert ^{q}bigr),dmu \ & {}- int _{V} F(x,lambda e_{1},lambda e_{2}) \ leq & frac{1}{p}lambda ^{p} Vert e_{1} Vert ^{p}_{W^{m_{1},p}(V)}+ frac{1}{q}lambda ^{q} Vert e_{2} Vert ^{q}_{W^{m_{2},q}(V)} \ & {}- biggl(frac{1}{p} frac{ Vert e_{1} Vert ^{p}_{W^{m_{1},p}(V)}}{ Vert e_{1} Vert ^{p}_{L^{p}(V)}}+ frac{1}{q} frac{ Vert e_{2} Vert ^{q}_{W^{m_{2},q}(V)}}{ Vert e_{2} Vert ^{q}_{L^{q}(V)}}+ frac{varepsilon _{1}}{2} biggr) bigl(lambda ^{p} Vert e_{1} Vert ^{p}_{L^{p}(V)}+ lambda ^{q} Vert e_{2} Vert ^{q}_{L^{q}(V)} bigr) \ &{}+C_{5}sum_{xin V}mu (x) \ leq & -frac{varepsilon _{1}}{2}lambda ^{p} Vert e_{1} Vert ^{p}_{L^{p}(V)},dmu – frac{varepsilon _{1}}{2}lambda ^{q} Vert e_{2} Vert ^{q}_{L^{q}(V)}+C_{5} sum _{xin V}mu (x) \ to & -infty ,quad text{as }lambda to infty . end{aligned}

Hence there exists a sufficiently large (lambda ^{*}>1 ) such that (varphi (lambda ^{*} e_{1},lambda ^{*} e_{2})<0). Let (lambda ^{*} e_{1}=u_{0}) and (lambda ^{*} e_{2}=v_{0}). Then (varphi (u_{0},v_{0})leq 0 ). □

### Proof of Theorem 1.1

It is easy to see that (varphi (0,0)=0). It follows from Lemmas 2.1 and 3.13.3, φ possesses a critical value (cge alpha >0), that is, there exists a point ((u_{*},v_{*})in W) such that

$$varphi (u_{*},v_{*})=cquad text{and}quad varphi ‘(u_{*},v_{*})=0.$$

Hence the associated point ((u_{*},v_{*})in W) is a nontrivial weak solution of system (1.1). □

### Lemma 3.4

Assume that ((F_{1})) and ((F_{3})) hold. Then for any finitedimensional subspace (widetilde{X}subset W), there is (R=R(widetilde{X})>0) such that (varphi (u)leq 0 ) on (widetilde{X}backslash B_{R}(0)).

### Proof

Let (operatorname{dim} widetilde{X}=m ). Then there exist positive constants (C_{6}(m)) and (C_{7}(m)) such that

$$Vert u Vert _{W^{m_{1},p}(V)}leq C_{6}(m) Vert u Vert _{L^{p}(V)},qquad Vert v Vert _{W^{m_{2},q}(V)} leq C_{7}(m) Vert v Vert _{L^{q}(V)}$$

(3.7)

for all ((u,v)in widetilde{X}). By ((F_{3})) we know that there exist constants (beta >frac{C_{6}(m)^{p}}{p}+frac{C_{7}(m)^{q}}{q}) and (r>0) such that

$$F(x,t,s)geq beta bigl( vert t vert ^{p}+ vert s vert ^{q}bigr)quad text{for all } biglvert (t,s) bigrvert geq r text{ and }xin V.$$

(3.8)

It follows from ((F_{1})) and (3.8) that there exists (C_{8}>0) such that

$$F(x,t,s)geq beta bigl( vert t vert ^{p}+ vert s vert ^{q}bigr)-C_{8} quad text{for all } (t,s) in mathbb{R}^{2} text{ and }xin V.$$

(3.9)

Then by (3.7) and (3.9) we have

begin{aligned}& varphi (u,v) \& quad = frac{1}{p} int _{V}bigl( biglvert nabla ^{m_{1}}u bigrvert ^{p}+h_{1}(x) vert u vert ^{p}bigr),dmu +frac{1}{q} int _{V}bigl( biglvert nabla ^{m_{2}}v bigrvert ^{q}+h_{2}(x) vert v vert ^{q}bigr),dmu – int _{V} F(x,u,v),dmu \& quad leq frac{1}{p} Vert u Vert ^{p}_{W^{m_{1},p}(V)}+ frac{1}{q} Vert v Vert ^{q}_{W^{m_{2},q}(V)}- beta bigl( Vert u Vert ^{p}_{L^{p}}+ Vert v Vert ^{q}_{L^{q}}bigr)+C_{8}sum _{xin V}mu (x) \& quad leq frac{1}{p} Vert u Vert ^{p}_{W^{m_{1},p}(V)}+ frac{1}{q} Vert v Vert ^{q}_{W^{m_{2},q}(V)} -beta biggl(frac{1}{C_{6}^{p}(m)} Vert u Vert ^{p}_{W^{m_{1},p}(V)}+ frac{1}{C_{7}^{q}(m)} Vert v Vert ^{q}_{W^{m_{2},q}(V)} biggr) \& qquad {}+C_{8}sum_{x in V}mu (x), end{aligned}

for all ((u,v)in widetilde{X}). Note that (beta >frac{C_{6}(m)^{p}}{p}+frac{C_{7}(m)^{q}}{q}). So (varphi (u,v)to -infty ) as (|(u,v)|to infty ). Thus we complete the proof. □

### Proof of Theorem 1.2

By ((F_{1})) and ((F_{5})) we know that φ is even and (varphi (0,0)=0). Let (X=W), (Y={0}) and (Z=W). Then by Lemma 3.1, Lemma 3.2, Lemma 3.4, Remark 2.1, Remark 2.2, and Lemma 2.2 we obtain that φ possesses at least dimW critical values. Thus we complete the proof. □