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Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type

  • Yongwen Liang EMAIL logo and Tianlan Chen
Published/Copyright: July 28, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

We study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator

Δ Δ u ( k − 1 ) 1 − ( Δ u ( k − 1 ) ) 2 + λ μ ( k ) ( p + 1 ) u p ( k ) v q + 1 ( k ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ Δ v ( k − 1 ) 1 − ( Δ v ( k − 1 ) ) 2 + λ μ ( k ) ( q + 1 ) u p + 1 ( k ) v q ( k ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) ,

where n ∈ N with n > 4 , max { p , q } > 1 , λ > 0 , Δ u ( k − 1 ) = u ( k ) − u ( k − 1 ) , and μ ( k ) > 0 for all k ∈ [ 2 , n − 1 ] Z . The existence of zero at least one or two positive solutions for the system are obtained according to the different intervals of λ . Our main tools are based on topological methods, critical point theory, and lower and upper solutions.

Keywords: discrete systems; Minkowski curvature operator; positive solution; Brouwer degree; lower and upper solutions; critical point theory
MSC 2010: 39A13; 47H11

1 Introduction

Set [ a , b ] Z ≔ { a , a + 1 , … , b } with a , b ∈ Z and a < b , where Z is the integer set. In this article, we show the nonexistence, existence, and multiplicity of solutions for the one-parameter discrete system

(1.1) Δ Δ u ( k − 1 ) 1 − ( Δ u ( k − 1 ) ) 2 + λ μ ( k ) ( p + 1 ) u p ( k ) v q + 1 ( k ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ Δ v ( k − 1 ) 1 − ( Δ v ( k − 1 ) ) 2 + λ μ ( k ) ( q + 1 ) u p + 1 ( k ) v q ( k ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) ,

where n ∈ N with n > 4 , p , q > 0 and max { p , q } > 1 , and μ ( k ) > 0 for all k ∈ [ 2 , n − 1 ] Z .

After some pioneering works by [1–3], there are lots of papers on the existence and multiplicity of positive solutions of the Dirichlet problem for the single differential equation with Minkowski curvature operator (see (1.1)), for example, see [4–13]. However, the study for systems with such an operator was recently initiated in [14], see also [15,16]. An important type of nonlinearities related to these studies is the Lane-Emden nonlinearities, see [17–20]. Gurban and Jebelean [16] established the non-existence and existence of solutions for differential systems

(1.2) div ∇ u 1 − ∣ ∇ u ∣ 2 + λ μ ( ∣ x ∣ ) ( p + 1 ) u p v q + 1 = 0 , in B ( R ) , div ∇ v 1 − ∣ ∇ v ∣ 2 + λ μ ( ∣ x ∣ ) ( q + 1 ) u p + 1 v q = 0 , in B ( R ) , u = 0 = v , on ∂ B ( R ) ,

where B ( R ) = { u ∈ R N : ∣ u ∣ < R } for some R > 0 , p , q > 0 with max { p , q } > 1 , λ > 0 is a parameter and μ : [ 0 , R ] → ( 0 , + ∞ ) is continuous, they developed the single equation in [6] to (1.2). At the same time, [21, Theorem 1.1] has proved the corresponding one of [6], that is, the difference problem

Δ Δ u ( k − 1 ) 1 − ( Δ u ( k − 1 ) ) 2 + λ μ ( k ) ( u ( k ) ) γ = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = 0 = u ( n ) ,

where γ > 1 .

Thus, it is natural to extend the aforementioned studies to discrete Lane-Emden systems (1.1) with Minkowski curvature operator. To wit, via the upper and lower solutions, Brouwer degree theory, and a crucial critical point theory, we have

Theorem 1.1

Assume n > 4 , p , q > 0 with max { p , q } > 1 and μ ( k ) > 0 for all k ∈ [ 2 , n − 1 ] Z . Then there exists Λ > 0 such that (1.1) has zero, at least one or two positive solutions according to λ ∈ ( 0 , Λ ) , λ = Λ , or λ > Λ .

Note that Theorem 1.1 extends the results of [16, Theorem 5.1] and [21, Theorem 1.1]. While the study of solutions for the discrete problem of prescribed mean curvature equations in Minkowski space is comparatively little, see [21–23].

The critical point theory [24] will be applied to prove the existence of solutions for a mixed boundary value problem involving a more general nonlinearity than that in (1.1). Actually, there exists the great challenge of a direct use of the critical point theory to discrete systems since a framework will be constructed to take a solution to boundary value problems as a critical point of a smooth functional defined on a finite dimensional space. Guo and Yu [25,26] first made the major breakthrough that the discrete Hamiltonian systems with variational structure were discussed. In addition, the multiplicity of solutions of difference systems without variational structure were proved by Balanov [27]. For more results concerning the discrete problems via the critical point theory, see [28–32], and for other results on nonlinear difference problems see, for instance, [33–38].

The rest of the article is organized as follows. Section 2 introduces some preliminaries. Lower and upper solution results are proved in Section 3. In Section 4, it is shown that there exists a nontrivial solution for the discrete systems involving the variational structure. Section 5 analyzes some important monotonicity properties of positive solutions and the Brouwer degree in zero of the fixed point operator introduced in Section 2 is 1. The multiplicity theorem will be proved in Section 6.

2 Preliminaries

Throughout this article, we assume

( H ϕ ) ϕ : ( − a , a ) → R ( 0 < a < ∞ ) is an odd, increasing homeomorphism.

For each p ∈ Z ( p ≥ 1 ) and u = ( u ( 1 ) , … , u ( p ) ) ∈ R p . Set ‖ u ‖ ∞ = max k ∈ [ 1 , p ] Z ∣ u ( k ) ∣ . Let α , β ∈ R p , we write α ≤ β (resp. α < β ) if α ( k ) ≤ β ( k ) (resp. α ( k ) < β ( k ) ) on k ∈ [ 1 , p ] Z .

Fix n > 4 with n ∈ N . Let

V n − 2 = { ( u , v ) ∈ R n × R n : Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) }

with the norm

‖ ( u , v ) ‖ = max { ‖ u ‖ ∞ , ‖ v ‖ ∞ } .

In this section, consider the general discrete system

(2.1) Δ [ ϕ ( Δ u ( k − 1 ) ) ] + f 1 ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ [ ϕ ( Δ v ( k − 1 ) ) ] + f 2 ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) ,

and suppose

(H1) f 1 , f 2 : [ 2 , n − 1 ] Z × R 2 → R are continuous functions.

Define the operators

S : R n − 2 → R n − 2 , Su ( k ) = ∑ j = 2 k u ( j ) , k ∈ [ 2 , n − 1 ] Z , K : R n − 2 → R n − 2 , Ku ( k ) = ∑ j = k n − 1 u ( j ) , k ∈ [ 2 , n − 1 ] Z .

It is not difficult to see that K ∘ ϕ − 1 ∘ S : R n − 2 → R n − 2 is continuous.

Denote by N f i the Nemytskii operator to f i ( i = 1 , 2 ),

N f i : R n → R n − 2 , N f i ( u ) = ( f i ( 2 , u ( 2 ) , v ( 2 ) ) , … , f i ( n − 1 , u ( n − 1 ) , v ( n − 1 ) ) ) .

Observe that ( u , v ) ∈ V n − 2 is a solution of (2.1) if and only if ( u , v ) ∈ V n − 2 is a fixed point of the continuous operator

N f : V n − 2 → V n − 2 , N f = ( K ∘ ϕ − 1 ∘ S ∘ N f 1 , K ∘ ϕ − 1 ∘ S ∘ N f 2 ) .

Lemma 2.1

Let ( u , v ) ∈ V n − 2 be a fixed point of N f . Then

(2.2) ‖ Δ u ‖ ∞ < a , ‖ Δ v ‖ ∞ < a , ‖ u ‖ ∞ < ( n − 1 ) a , ‖ v ‖ ∞ < ( n − 1 ) a ,

and hence,

(2.3) deg [ I − N f , B ρ , 0 ] = 1 , for a l l ρ ≥ a n .

Moreover, (2.1) always has a solution in B ρ for all ρ ≥ a n .

Proof

It is apparent that (2.2) holds according to the range of ϕ − 1 .

Now let us consider the compact homotopy

ℋ : [ 0 , 1 ] × V n − 2 → V n − 2 , ℋ ( τ , ( ⋅ , ⋅ ) ) = τ N f ( ⋅ , ⋅ ) .

(2.2) implies ℋ ( [ 0 , 1 ] × V n − 2 ) ⊂ B a n . Combining this with the invariance under homotopy of the Brouwer degree, we can deduce

deg [ I − ℋ ( 0 , ( ⋅ , ⋅ ) ) , B ρ , 0 ] = deg [ I − ℋ ( 1 , ( ⋅ , ⋅ ) ) , B ρ , 0 ] , for all ρ ≥ n a .

Since ℋ ( 0 , ( ⋅ , ⋅ ) ) = 0 , ℋ ( 1 , ( ⋅ , ⋅ ) ) = N f , and deg [ I , B ρ , 0 ] = 1 , then (2.3) is valid. Consequently, (2.1) always has a solution in B ρ for all ρ ≥ a n .□

3 Lower and upper solutions

Definition 3.1

A lower solution of (2.1) is a couple of functions ( α u , α v ) ∈ R n × R n satisfying ‖ Δ α u ‖ ∞ < a , ‖ Δ α v ‖ ∞ < a , and

(3.1) Δ [ ϕ ( Δ α u ( k − 1 ) ) ] + f 1 ( k , α u ( k ) , α v ( k ) ) ≥ 0 , k ∈ [ 2 , n − 1 ] Z , Δ [ ϕ ( Δ α v ( k − 1 ) ) ] + f 2 ( k , α u ( k ) , α v ( k ) ) ≥ 0 , k ∈ [ 2 , n − 1 ] Z , Δ α u ( 1 ) = 0 = Δ α v ( 1 ) , α u ( n ) ≤ 0 , α v ( n ) ≤ 0 .

Similarly, an upper solution ( β u , β v ) ∈ R n × R n of (2.1) is defined by reversing the aforementioned inequalities.

Proposition 3.1

Assume (H1), and for each i = 1 , 2 ,

(H2) f i ( k , s , t ) is quasi-monotone nondecreasing with respect to both s and t , i.e.,

f i ( k , s , t 1 ) ≤ f i ( k , s , t 2 ) as t 1 ≤ t 2 ( resp. f i ( k , s 1 , t ) ≤ f i ( k , s 2 , t ) as s 1 ≤ s 2 ) .

If (2.1) has a lower solution ( α u , α v ) and an upper solution ( β u , β v ) such that α u ≤ β u , α v ≤ β v , then there exists a solution ( u , v ) of (2.1) such that α u ≤ u ≤ β u and α v ≤ v ≤ β v .

Proof

Define Γ 1 , Γ 2 : [ 2 , n − 1 ] Z × R 2 → R ,

Γ 1 ( k , s , t ) = f 1 ( k , γ 1 ( k , s ) , γ 2 ( k , t ) ) − s + γ 1 ( k , s ) , Γ 2 ( k , s , t ) = f 2 ( k , γ 1 ( k , s ) , γ 2 ( k , t ) ) − t + γ 2 ( k , t ) ,

where γ i is given by

γ 1 ( k , ξ ) = max { α u ( k ) , min { ξ , β u ( k ) } } , γ 2 ( k , ξ ) = max { α v ( k ) , min { ξ , β v ( k ) } } .

Consider the auxiliary problem

(3.2) Δ [ ϕ ( Δ u ( k − 1 ) ) ] + Γ 1 ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ [ ϕ ( Δ v ( k − 1 ) ) ] + Γ 2 ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) .

Using the same process argument to treat [21, Proposition 2.1], it can be easily deduced that (3.2) has at least one solution.

We will prove that if ( u , v ) is a solution of (3.2), then

α u ≤ u ≤ β u and α v ≤ v ≤ β v .

We only need to show that α u ≤ u , the other parts can be proved similarly.

Suppose, by contradiction, that there exists i ∈ [ 1 , n − 1 ] Z such that

(3.3) max k ∈ [ 2 , n − 1 ] Z ( α u ( k ) − u ( k ) ) = α u ( i ) − u ( i ) > 0 .

If i ∈ [ 2 , n − 1 ] Z , then Δ α u ( i ) ≤ Δ u ( i ) and Δ α u ( i − 1 ) ≥ Δ u ( i − 1 ) , and so

Δ [ ϕ ( Δ α u ( i − 1 ) ) ] ≤ Δ [ ϕ ( Δ u ( i − 1 ) ) ] .

Since ( α u , α v ) is a lower solution of (2.1) and (H2) holds, one has

Δ [ ϕ ( Δ α u ( i − 1 ) ) ] ≤ Δ [ ϕ ( Δ u ( i − 1 ) ) ] = − f 1 ( i , α u ( i ) , γ 2 ( i , v ( i ) ) ) + u ( i ) − α u ( i ) < − f 1 ( i , α u ( i ) , γ 2 ( i , v ( i ) ) ) ≤ − f 1 ( i , α u ( i ) , α v ( i ) ) ≤ Δ [ ϕ ( Δ α u ( i − 1 ) ) ] ,

which is a contradiction.

If i = n , then α u ( n ) > u ( n ) , but u ( n ) = 0 and α u ( n ) ≤ 0 , which is a contradiction.

If i = 1 , it follows from Δ α u ( 1 ) = 0 = Δ u ( 1 ) that α u ( 1 ) = α u ( 2 ) and u ( 1 ) = u ( 2 ) , then i = 2 is also a maximum point, by using the same process to treat the maximum point i ∈ [ 2 , n − 1 ] Z , the same contradiction is obtained.

Consequently, α u ( k ) ≤ u ( k ) for all k ∈ [ 1 , n ] Z .□

Lemma 3.1

Suppose that (2.1) has a lower solution ( α u , α v ) and an upper solution ( β u , β v ) satisfying α u ≤ β u , α v ≤ β v . Set

Ω ( α u , α v ) , ( β u , β v ) = { ( u , v ) ∈ V n − 2 : α u ≤ u ≤ β u , α v ≤ v ≤ β v } .

If (2.1) has a unique solution ( u 0 , v 0 ) ∈ Ω ( α u , α v ) , ( β u , β v ) and there exists ρ 0 > 0 such that B ¯ ( ( u 0 , v 0 ) , ρ 0 ) ⊂ Ω α , β , then

(3.4) deg [ I − N f , B ( ( u 0 , v 0 ) , ρ ) , 0 ] = 1 , for 0 < ρ ≤ ρ 0 .

Proof

Let N Γ be the fixed point operator of (3.2). By virtue of Proposition 3.1, if ( u , v ) is a fixed point of N Γ , then ( u , v ) ∈ Ω ( α u , α v ) , ( β u , β v ) , and also ( u , v ) is a fixed point of N f . Since ( u 0 , v 0 ) is a unique fixed point of N f , according to Lemma 2.1 and the excision property of the Brouwer degree, we can prove

deg [ I − N Γ , B ( ( u 0 , v 0 ) , ρ ) , 0 ] = 1 , for all ρ > 0 .

On account of N Γ ( u , v ) = N f ( u , v ) for ( u , v ) ∈ B ¯ ( ( u 0 , v 0 ) , ρ 0 ) , and so (3.4) is valid.□

4 Variational solutions

In this section, let us study the variational structure problems for (2.1),

(4.1) Δ [ ϕ ( Δ u ( k − 1 ) ) ] + F u ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ [ ϕ ( Δ v ( k − 1 ) ) ] + F v ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) .

Moreover, assume

( H Φ ) There exists a continuous function Φ : [ − a , a ] → R , Φ ( 0 ) = 0 , and Φ ∈ C 1 ( − a , a ) , let ϕ ≔ Φ ′ : ( − a , a ) → R be an increasing homeomorphism with ϕ ( 0 ) = 0 .

( H F ) There is a continuous function F = F ( k , u , v ) : [ 2 , n − 1 ] Z × R 2 → R such that F u and F v are continuous on [ 2 , n − 1 ] Z × R 2 .

Next we show the existence of nontrivial solutions of (4.1) via the critical point theory [24], for this, define the closed convex set K in R n ,

K ≔ { u ∈ R n : ‖ Δ u ‖ ∞ ≤ a } ,

which implies that K 0 ≔ { u ∈ K : u ( n ) = 0 } is also a convex, closed subset of R n . Owing to

(4.2) ‖ u ‖ ∞ ≤ ( n − 1 ) a , for all u ∈ K ,

therefore, K 0 is bounded in R n . So, we have that K 0 × K 0 ⊂ R n × R n is closed and convex.

Now let us introduce the functional ψ : R n → ( − ∞ , + ∞ ] ,

ψ ( u ) = ∑ k = 2 n − 1 Φ ( Δ u ( k ) ) , if u ∈ K 0 , + ∞ , if u ∈ R n \ K 0

is proper, convex, and lower semicontinuous. Note that ψ is bounded on K 0 . Denote Ψ : R n × R n → ( − ∞ , + ∞ ] by

Ψ ( u , v ) = ψ ( u ) + ψ ( v ) for all ( u , v ) ∈ R n × R n .

Then Ψ is also proper, convex, and lower semicontinuous.

Define the mapping ℱ : R n × R n → R by

ℱ ( u , v ) = ∑ k = 2 n − 1 F ( k , u ( k ) , v ( k ) ) , for ( u , v ) ∈ R n × R n .

It can be easily seen that ℱ ∈ C 1 ( R n × R n , R ) , and, for all u , v , w 1 , w 2 ∈ R n ,

⟨ ℱ ′ ( u , v ) , ( w 1 , w 2 ) ⟩ = ∑ k = 2 n − 1 [ F u ( k , u ( k ) , v ( k ) ) w 1 ( k ) + F v ( k , u ( k ) , v ( k ) ) w 2 ( k ) ] .

Setting

I ( u , v ) ≔ Ψ ( u , v ) − ℱ ( u , v ) , for any ( u , v ) ∈ R n × R n .

It is obvious that I is a Szulkin’s functional (see [30]). Accordingly, ( u , v ) ∈ R n × R n is a critical point of I if ( u , v ) ∈ K 0 × K 0 and

(4.3) Ψ ( w 1 , w 2 ) − Ψ ( u , v ) − ⟨ ℱ ′ ( u , v ) , ( w 1 − u , w 2 − v ) ⟩ ≥ 0 , for all w 1 , w 2 ∈ R n .

Proposition 4.1

Assume ( H Φ ) and ( H F ). Then, each critical point of I is a solution of (4.1). Moreover, (4.1) has a solution which is a minimum point of I on R n × R n .

Proof

Let ( u , v ) ∈ K 0 × K 0 be a critical point of I . Then ( u , v ) solves the variational inequality (4.3) with w 2 = v , we have

ψ ( w 1 ) − ψ ( u ) − ∑ k = 2 n − 1 F u ( k , u ( k ) , v ( k ) ) ( w 1 ( k ) − u ( k ) ) ≥ 0 , for all w 1 ∈ R n ,

which implies that u ∈ K 0 is a critical point of ψ ( ⋅ ) − ℱ ( ⋅ , v ) , from [21, Proposition 5.1], it follows

Δ [ ϕ ( Δ u ( k − 1 ) ) ] + F u ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 .

Similarly, we can show that v satisfies

Δ [ ϕ ( Δ v ( k − 1 ) ) ] + F v ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ v ( 1 ) = v ( n ) = 0 .

The rest of the proof is immediately proved by virtue of [21, Proposition 5.1].□

5 Sublinear growth at zero

In this section, we study the discrete systems

(5.1) Δ [ ϕ ( Δ u ( k − 1 ) ) ] + g 1 ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ [ ϕ ( Δ v ( k − 1 ) ) ] + g 2 ( k , u ( k ) , v ( k ) ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) .

Assume

( H g ) g 1 , g 2 : [ 2 , n − 1 ] Z × [ 0 , ∞ ) × [ 0 , ∞ ) → [ 0 , ∞ ) are continuous functions, and

  1. g 1 ( k , s , t ) > 0 < g 2 ( k , s , t ) for all ( k , s , t ) ∈ [ 2 , n − 1 ] Z × ( 0 , ∞ ) × ( 0 , ∞ ) ;

  2. g 1 ( k , ξ , 0 ) = 0 = g 2 ( k , 0 , ξ ) for all k ∈ [ 2 , n − 1 ] Z and ξ > 0 .

The next result provides us with some important monotonicity properties of solutions.

Lemma 5.1

Assume ( H ϕ ) and ( H g ). If ( u , v ) ∈ R n × R n is a nontrivial solution of

(5.2) Δ [ ϕ ( Δ u ( k − 1 ) ) ] + g 1 ( k , ∣ u ( k ) ∣ , ∣ v ( k ) ∣ ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ [ ϕ ( Δ v ( k − 1 ) ) ] + g 2 ( k , ∣ u ( k ) ∣ , ∣ v ( k ) ∣ ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) ,

then u ( k ) > 0 and v ( k ) > 0 on k ∈ [ 1 , n − 1 ] Z . Moreover, u ( k ) and v ( k ) are strictly decreasing on k ∈ [ 2 , n ] Z .

Proof

It follows from the first equation of (5.2) that

(5.3) ϕ [ Δ u ( k ) ] = − ∑ i = 2 k g 1 ( i , ∣ u ( i ) ∣ , ∣ v ( i ) ∣ ) ,

from ( H g ), one obtains Δ u ( k ) ≤ 0 , and hence, u ( k ) is decreasing on k ∈ [ 1 , n ] Z . Similarly, we can obtain that v ( k ) is decreasing on k ∈ [ 1 , n ] Z .

Since u ( n ) = 0 , then u ( k ) ≥ 0 on k ∈ [ 1 , n ] Z , and analogously v ( k ) ≥ 0 on k ∈ [ 1 , n ] Z . For any k ∈ [ 1 , n ] Z , assuming v ( k ) ≡ 0 , then u ( k ) ≢ 0 (note that ( u , v ) ∈ R n × R n is a nontrivial solution), which together with (5.3), ( H g )(ii) and u ( n ) = 0 imply u ( k ) ≡ 0 , we obtain a contradiction with u ( k ) ≢ 0 , and so

(5.4) v ( k ) ≢ 0 on k ∈ [ 1 , n ] Z .

A similar argument shows that

(5.5) u ( k ) ≢ 0 on k ∈ [ 1 , n ] Z .

From (5.4), (5.5) as well as u ( k ) and v ( k ) are decreasing, one has u ( 1 ) > 0 < v ( 1 ) , which together with (5.3) and ( H g )(i), we can prove that Δ u ( k ) < 0 for k ∈ [ 2 , n − 1 ] Z , i.e., u ( k ) is strictly decreasing with respect to k and u ( k ) > 0 for k ∈ [ 1 , n − 1 ] Z . Similarly, v ( k ) is strictly decreasing with respect to k and v ( k ) > 0 for k ∈ [ 1 , n − 1 ] Z .□

According to Lemma 5.1, it can be easily seen that ( u , v ) is a nontrivial solution of (5.2) if and only if ( u , v ) is a positive solution of (5.1).

Lemma 5.2

Assume ( H ϕ ) and ( H g ) hold, and there is some M > 0 such that either

(5.6) lim s → 0 + g 1 ( k , s , t ) s = 0 uniformly o n k ∈ [ 2 , n − 1 ] Z , t ∈ [ 0 , M ] ,

or

(5.7) lim t → 0 + g 2 ( k , s , t ) t = 0 uniformly o n k ∈ [ 2 , n − 1 ] Z , s ∈ [ 0 , M ] .

Then there exists ρ 0 > 0 such that

deg [ I − N g , B ρ , 0 ] = 1 for a l l 0 < ρ ≤ ρ 0 ,

where N g is the fixed point operator to (5.1).

Proof

Let

(5.8) 0 < ε < 1 ( n − 2 ) 2 .

We only show the case (5.6) (the similar result is true for the case (5.7)). Then there exists s ε > 0 such that, for all s ∈ ( 0 , s ε ) ,

(5.9) g 1 ( k , s , t ) ≤ ε ϕ ( s ) , for k ∈ [ 2 , n − 1 ] Z , t ∈ [ 0 , M ] .

Consider the compact homotopy

ℋ : [ 0 , 1 ] × V n − 2 → V n − 2 , ℋ ( τ , u , v ) = τ N g ( u , v ) .

We claim that there exists ρ 0 > 0 such that

(5.10) ℋ ( τ , u , v ) ≠ ( u , v ) for all ( τ , u , v ) ∈ [ 0 , 1 ] × ( B ¯ ρ 0 \ { ( 0 , 0 ) } ) .

Suppose on the contrary that there exist a sequence of

{ τ m } ⊂ [ 0 , 1 ] , { ( u m , v m ) } ⊂ V n − 2 \ { ( 0 , 0 ) } for all m ∈ N and ‖ ( u m , v m ) ‖ → 0 ,

such that

( u m , v m ) = τ m N g ( u m , v m ) .

It follows from Lemma 5.1 that u m ( k ) > 0 and v m ( k ) > 0 on k ∈ [ 1 , n − 1 ] Z . We may assume (passing if necessary to a subsequence) that

‖ u m ‖ ∞ ≤ s ε , ‖ v m ‖ ∞ ≤ M , for all m ∈ N .

On account of (5.9), we have

g 1 ( k , u m ( k ) , u m ( k ) ) ≤ ε ϕ ( ‖ u m ‖ ∞ ) , for k ∈ [ 2 , n − 1 ] Z , m ∈ N .

Thus, for any m ∈ N , one obtains

‖ u m ‖ ∞ ≤ ∑ k = 2 n − 1 ϕ − 1 ∑ i = 2 k g 1 ( i , u m ( i ) , v m ( i ) ) ≤ ∑ k = 2 n − 1 ϕ − 1 ∑ i = 2 k ε ϕ ( ‖ u m ‖ ∞ ) ≤ ( n − 2 ) ϕ − 1 [ ( n − 2 ) ε ϕ ( ‖ u m ‖ ∞ ) ] ,

and, hence,

ϕ ‖ u m ‖ ∞ n − 2 ϕ ( ‖ u m ‖ ∞ ) ≤ ( n − 2 ) ε , for all m ∈ N .

When m → ∞ , it can be seen 1 n − 2 ≤ ( n − 2 ) ε , which contradicts with (5.8). Thus, (5.10) is valid. Using the invariance under homotopy of the Brouwer degree, we have that

□ deg [ I − N g , B ρ , 0 ] = deg ( I , B ( ρ ) , 0 ) = 1 , for all ρ ∈ ( 0 , ρ 0 ] .

6 Proof of main result

In this section, we study the one-parameter discrete system (2.1). Set

ϕ ( s ) = s 1 − s 2 , s ∈ ( − 1 , 1 ) .

Note that ( u , v ) ∈ R n × R n is a positive solution of (2.1) if and only if it is a nontrivial solution

(6.1) Δ Δ u ( k − 1 ) 1 − ( Δ u ( k − 1 ) ) 2 + λ μ ( k ) ( p + 1 ) ∣ u ( k ) ∣ p ∣ v ( k ) ∣ q + 1 = 0 , k ∈ [ 2 , n − 1 ] Z , Δ Δ v ( k − 1 ) 1 − ( Δ v ( k − 1 ) ) 2 + λ μ ( k ) ( q + 1 ) ∣ u ( k ) ∣ p + 1 ∣ v ( k ) ∣ q = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) .

In such circumstances, u ( k ) and v ( k ) are strictly decreasing about k .

Now, we give the proof of the main result (Theorem 1.1) in this article.

Proof of Theorem 1.1

We assume that p > 1 , q > 0 . Let

S j = { λ > 0 : ( 2.1 ) has at least j positive solutions } = { λ > 0 : ( 6.1 ) has at least j nontrivial solutions } ( j = 1 , 2 ) .

Step 1: Existence of Λ . Let ( u , v ) be a positive solution of (2.1) for some λ > 0 . Summing the first equation in (2.1) from 2 to k − 1 , we have

− Δ u k − 1 1 − ( Δ u k − 1 ) 2 = λ ( p + 1 ) ∑ i = 2 k − 1 μ ( i ) u p ( i ) v q + 1 ( i ) , k ∈ [ 2 , n − 1 ] Z .

Since u ( k ) and v ( k ) are strictly decreasing on k ∈ [ 2 , n ] Z , then, for any k ∈ [ 2 , n ] Z , it can be easily seen that

− Δ u k − 1 ≤ − Δ u k − 1 1 − ( Δ u k − 1 ) 2 ≤ λ ( p + 1 ) u p ( 1 ) v q + 1 ( 1 ) μ ¯ ( k − 2 )

(where μ ¯ = max k ∈ [ 2 , n − 1 ] Z μ ( k ) ) and summing over [ 2 , n ] Z , one obtains

(6.2) u ( 1 ) ≤ λ ( p + 1 ) u p ( 1 ) v q + 1 ( 1 ) μ ¯ ( n − 1 ) ( n − 2 ) 2 < λ ( p + 1 ) u p ( 1 ) v q + 1 ( 1 ) μ ¯ ( n − 1 ) 2 2 .

By virtue of 0 < u ( 1 ) , v ( 1 ) < n − 1 (see (2.2)) and p > 1 , we have that

(6.3) λ > 2 ( p + 1 ) μ ¯ ( n − 1 ) p + q + 2 .

Define the functional I λ : R n × R n → ( 0 , + ∞ ] to (6.1) by

I λ ( u , v ) = 2 ( n − 2 ) − ∑ k = 2 n − 1 [ 1 − ( Δ u ( k ) ) 2 + 1 − ( Δ v ( k ) ) 2 ] − 2 λ ∑ k = 2 n − 1 μ ( k ) ∣ u ( k ) ∣ p + 1 ∣ v ( k ) ∣ q + 1

for ( u , v ) ∈ K 0 × K 0 . It can be deduced that I λ = + ∞ on R n × R n \ ( K 0 × K 0 ) . Let u 0 ( k ) = n − k = v 0 ( k ) on k ∈ [ 1 , n ] Z . Then, for the sufficiently large λ > 0 , I λ ( u 0 , v 0 ) < 0 . Hence, for such λ , the functional I λ has a negative minimum, it follows from Proposition 4.1 and I λ ( 0 , 0 ) = 0 that system (6.1) has a nontrivial solution. In particular, S 1 ≠ ∅ . Then, setting

Λ = Λ ( n ) ≔ inf S 1 ( < + ∞ ) ,

it is apparent that Λ ≥ 2 ( p + 1 ) μ ¯ ( n − 1 ) p + q + 2 .

We claim that Λ ∈ S 1 . For this, let { λ m } ⊂ S 1 satisfy λ m → Λ and { ( u m , v m ) } ⊂ V n − 2 with

(6.4) u m ( k ) > 0 < v m ( k ) on k ∈ [ 1 , n − 1 ] Z ,

(6.5) u m ( k ) = K ∘ ϕ − 1 ∘ S ∘ [ λ m ( p + 1 ) μ ( k ) u m p ( k ) v m q + 1 ( k ) ] ,

(6.6) v m ( k ) = K ∘ ϕ − 1 ∘ S ∘ [ λ m ( q + 1 ) μ ( k ) u m p + 1 ( k ) v m q ( k ) ] .

Using the Arzela-Ascali theorem and (2.2), it can be seen that there exists ( u , v ) ∈ R n × R n , after taking a subsequence if necessary, such that { ( u m , v m ) } converges to ( u , v ) in R n × R n . Thus, from (6.4)–(6.6), we have u ( k ) ≥ 0 ≤ v ( k ) on k ∈ [ 1 , n − 1 ] Z , and

u ( k ) = K ∘ ϕ − 1 ∘ S ∘ [ Λ ( p + 1 ) μ ( k ) u p ( k ) v q + 1 ( k ) ] , v ( k ) = K ∘ ϕ − 1 ∘ S ∘ [ Λ ( q + 1 ) μ ( k ) u p + 1 ( k ) v q ( k ) ] .

From (6.2), we obtain that there is c > 0 such that u m ( 1 ) > c for all m sufficiently large, which implies that u ( 1 ) ≥ c . This and Lemma 5.1 give u ( k ) > 0 < v ( k ) on [ 1 , n − 1 ] Z . Consequently, Λ ∈ S 1 and

Λ > 2 ( p + 1 ) μ ¯ ( n − 1 ) p + q + 2 .

Next, for any λ 0 > Λ , we will show λ 0 ∈ S 1 . Suppose ( u 1 , v 1 ) is a positive solution of (2.1) with λ = Λ . It can be easily seen that ( u 1 , v 1 ) is a lower solution of (6.1) with λ = λ 0 . To construct an upper solution of (6.1) with λ = λ 0 , choose H 1 > 0 < H 2 and n ˜ > n + 1 , it is not difficult to prove that

(6.7) Δ Δ u ( k − 1 ) 1 − ( Δ u ( k − 1 ) ) 2 + H 1 = 0 , k ∈ [ 2 , n ˜ − 1 ] Z , Δ Δ v ( k − 1 ) 1 − ( Δ v ( k − 1 ) ) 2 + H 2 = 0 , k ∈ [ 2 , n ˜ − 1 ] Z , Δ u ( 1 ) = u ( n ˜ ) = 0 = Δ v ( 1 ) = v ( n ˜ )

has the unique (positive) solution

(6.8) u 2 ( k ) = ∑ j = k n ˜ − 1 H 1 ( j − 1 ) 1 + H 1 2 ( j − 1 ) 2 , k ∈ [ 2 , n ˜ − 1 ] Z ,

(6.9) v 2 ( k ) = ∑ j = k n ˜ − 1 H 2 ( j − 1 ) 1 + H 2 2 ( j − 1 ) 2 , k ∈ [ 2 , n ˜ − 1 ] Z .

Let us fix λ 2 > λ 0 and

H 1 = λ 2 ( p + 1 ) μ ¯ ( n ˜ − 1 ) p + q + 1 , H 2 = λ 2 ( q + 1 ) μ ¯ ( n ˜ − 1 ) p + q + 1 .

Owing to u 2 ( n ) > 0 < v 2 ( n ) and

λ 0 ( p + 1 ) μ ( k ) u 2 p ( k ) v 2 q + 1 ( k ) ≤ λ 2 ( p + 1 ) μ ¯ ( n ˜ − 1 ) p + q + 1 , λ 0 ( q + 1 ) μ ( k ) u 2 p + 1 ( k ) v 2 q ( k ) ≤ λ 2 ( q + 1 ) μ ¯ ( n ˜ − 1 ) p + q + 1 ,

therefore, ( u 2 , v 2 ) is an upper solution of (6.1) with λ = λ 0 . Observe that

u 2 ( n ) = H 1 ∑ j = n n ˜ − 1 j − 1 1 + H 1 2 ( j − 1 ) 2 = H 1 ∑ j = n − 1 n ˜ − 2 j 1 + H 1 2 j 2 ,

then, we can find a sufficiently large n ˜ such that u 1 ( 1 ) < u 2 ( n ) . Similarly, we can obtain v 1 ( 1 ) < v 2 ( n ) . Combining this with u 1 ( k ) , v 1 ( k ) , u 2 ( k ) , and v 2 ( k ) are strictly decreasing, we can deduce that u 1 ( k ) < u 2 ( k ) and v 1 ( k ) < v 2 ( k ) for all k ∈ [ 1 , n ] Z . Therefore, by Proposition 3.1, λ 0 ∈ S 1 , i.e., S 1 = [ Λ , ∞ ) .

Step 2: Multiplicity. Fix λ 0 > Λ . We will show that λ 0 ∈ S 2 . Suppose that ( u 1 , v 1 ) is a lower solution and ( u 2 , v 2 ) is an upper solution constructed as in step 1. Let ( u 0 , v 0 ) be a solution of (6.1) with λ = λ 0 such that ( u 0 , v 0 ) ∈ Ω ≔ Ω ( u 1 , v 1 ) , ( u 2 , v 2 ) (see Lemma 3.2).

First, we claim that there exists ε > 0 such that B ¯ ( ( u 0 , v 0 ) , ε ) ⊂ Ω . It follows from (6.8) that

u 2 ( k ) = ∑ i = k n ˜ − 1 ϕ − 1 ∑ j = 2 i λ 2 ( p + 1 ) μ ¯ ( n ˜ − 1 ) p + q + 1 , for all k ∈ [ 2 , n − 1 ] Z ,

and hence,

u 2 ( k ) > ∑ i = k n − 1 ϕ − 1 ∑ j = 2 i λ 2 ( p + 1 ) μ ¯ ( n ˜ − 1 ) p + q + 1 ≥ ∑ i = k n − 1 ϕ − 1 ∑ j = 2 i λ 0 ( p + 1 ) μ ( j ) u 0 p ( j ) v 0 q + 1 ( j ) = u 0 ( k ) .

Analogously, we obtain that v 2 ( k ) > v 0 ( k ) . Therefore, there exists ε 2 > 0 such that

‖ u − u 0 ‖ ≤ ε 2 ⇒ u ≤ u 2 and ‖ v − v 0 ‖ ≤ ε 2 ⇒ v ≤ v 2 , for ( u , v ) ∈ V n − 2 .

Using the similar argument we have u 1 < u 0 and v 1 < v 0 on [ 1 , m ] Z for some m ∈ [ 2 , n − 1 ] Z . Thus, we can find ε 3 > 0 such that, for k ∈ [ 1 , m ] Z ,

(6.10) ‖ u − u 0 ‖ ≤ ε 3 ⇒ u 1 ≤ u and ‖ v − v 0 ‖ ≤ ε 3 ⇒ v 1 ≤ v , for ( u , v ) ∈ V n − 2 .

If m = n − 1 , the claim is valid. Otherwise, it follows from

− Δ u 0 ( k ) = ϕ − 1 ∘ S ∘ λ 0 μ ( k ) u 0 p ( k ) v 0 q + 1 ( k ) and − Δ u 1 ( k ) = ϕ − 1 ∘ S ∘ Λ μ ( k ) u 1 p ( k ) v 1 q + 1 ( k )

that Δ u 0 ( k ) < Δ u 1 ( k ) on [ m , n − 1 ] Z . Analogously, we obtain that Δ v 0 ( k ) < Δ v 1 ( k ) on [ m , n − 1 ] Z . So, choose the sufficiently small ε 1 ∈ ( 0 , ε 3 ) such that Δ u ( k ) < Δ u 1 ( k ) and Δ v ( k ) < Δ v 1 ( k ) on [ m , n − 1 ] Z if ( u , v ) ∈ B ( ( u 0 , v 0 ) , ε 1 ) . Then, using u 0 ( n ) = 0 = u ( n ) and v 0 ( n ) = 0 = v ( n ) , we deduce that u > u 1 and v > v 1 on [ m , n − 1 ] Z for all ( u , v ) ∈ B ¯ ( ( u 0 , v 0 ) , ε 1 ) . Now, on account of (6.10), the claim is valid for any ε ∈ ( 0 , min { ε 1 , ε 2 } ) .

Next, if (6.1) has a second solution ( u , v ) ∈ Ω ( u 1 , v 1 ) , ( u 2 , v 2 ) , which is nontrivial and so S 2 ≠ ∅ . Otherwise, from Lemma 3.2, we obtain

deg [ I − N λ 0 , B ( ( u 0 , v 0 ) , ρ ) , 0 ] = 1 , for all 0 < ρ ≤ ε ,

where N λ 0 is the fixed point operator to (6.1) with λ = λ 0 . Moreover, by virtue of Lemma 2.1, we have

deg [ I − N λ 0 , B ρ , 0 ] = 1 , for all ρ ≥ n ,

and, by Lemma 5.2, one obtains

deg [ I − N λ 0 , B ρ , 0 ] = 1 , for sufficiently small ρ .

Now, taking ρ 3 ≥ n and sufficiently small ρ 1 , ρ 2 > 0 such that

B ¯ ( ( u 0 , v 0 ) , ρ 1 ) ∩ B ¯ ρ 2 = ∅ and B ¯ ( ( u 0 , v 0 ) , ρ 1 ) ∪ B ¯ ρ 2 ⊂ B ρ 3 .

Thus, using the additivity-excision property of the Brouwer degree, it follows

deg [ I − N λ 0 , B ρ 3 \ [ B ¯ ( ( u 0 , v 0 ) , ρ 1 ) ∪ B ¯ ρ 2 ] , 0 ] = − 1 ,

which implies that N λ 0 has a fixed point ( u ˜ 0 , v ˜ 0 ) ∈ B ρ 3 \ [ B ¯ ( ( u 0 , v 0 ) , ρ 1 ) ∪ B ¯ ρ 2 ] , i.e., (2.1) has a second positive solution.□

Acknowledgements

Not applicable.

  1. Funding information: This work was supported by the NSFC (No. 11901464, 11801453, 12061064), Gansu Provincial National Science Foundation of China (No. 20JR10RA100, 21JR1RA230), Department of EducationUniversity Innovation Fund of Gansu Provincial (Grant No. 2022A-218, 2021A-006).

  2. Author contributions: YL carried out the analysis and proof of the main results and was a major contributor in writing the manuscript. TC participated in checking the proofs and checked typing errors in the text. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-06-24
Revised: 2023-03-09
Accepted: 2023-06-13
Published Online: 2023-07-28

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
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  122. Approximate solvability method for nonlocal impulsive evolution equation
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  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
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  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2022-0602
Keywords for this article
discrete systems; Minkowski curvature operator; positive solution; Brouwer degree; lower and upper solutions; critical point theory
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