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Home Existence and properties of soliton solution for the quasilinear Schrödinger system
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Existence and properties of soliton solution for the quasilinear Schrödinger system

  • Xue Zhang and Jing Zhang EMAIL logo
Published/Copyright: July 22, 2024

Abstract

In this article, we consider the following quasilinear Schrödinger system:

{εΔu+u+k2ε[Δu2]u=2αα+βuα2uvβ,xRN,εΔv+v+k2ε[Δv2]v=2βα+βuαvβ2v,xRN,

where ε>0,k<0 are real constants, N3 , α,β are integers multiple of constant 2. By using the Mountain Pass Theorem in a suitable Orlicz space proposed by Abbas Moameni [Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN , J. Differential Equations 229 (2006), 570–587], we proved the existence of soliton solution (uε,vε) for the above system, and (uε(x),vε(x))(0,0) as ε0 .

MSC 2010: 35J10; 35J50; 35C08

1 Introduction

We consider the following quasilinear Schrödinger system:

(1.1) {εΔu+u+k2ε[Δu2]u=2αα+βuα2uvβ,xRN,εΔv+v+k2ε[Δv2]v=2βα+βuαvβ2v,xRN,

where N3 , ε>0,k<0 are the real constants.

In recent years, much more attention has been devoted to the quasilinear Schrödinger system

(1.2) {Δu+V1(x)u+k2[Δu2]u=λf(x,u,v),xRN,Δv+V2(x)v+k2[Δv2]v=λh(x,u,v),xRN.

Li proved the existence of nontrivial solution by using a change of variable and the Mountain Pass Theorem when k>0 is large enough [1]. Let V1(x)=λa(x)+1,V2(x)=λb(x)+1,k=1 and λf(x,u,v)=2αα+βuα2uvβ,λh(x,u,v)=2βα+βuαvβ2v , Guo and Tang proved the existence of ground state solution by using the Nehari manifold method and concentration compactness principle in [2], which is localized near the potential well int{a1(0)}=int{b1(0)} for λ large enough. In [3], minimax methods in a suitable Orlicz space were employed to establish the existence of standing wave solution for a quasilinear Schrödinger system involving subcritical nonlinearities.

For some systems similar to (1.2), Chen and Zhang have done several contributions. In [4], they obtained the existence of positive ground state solution by using Morse iteration to define a Pohožaev manifold. They obtained the existence of positive solution by using monotonicity trick and the Morse iteration in [5]. They proved the existence of ground state solution by minimization under a convenient constraint and concentration compactness lemma in [6]. They found the existence of ground state solution by establishing a suitable constraint set and studying related minimization problem in [7].

By establishing a suitable Nehari-Pohožaev-type constraint set and considering related minimization problem, the existence of ground state solution for a class of systems was proved in [8]. The symmetric Mountain Pass Theorem was employed to establish the existence of infinitely many solutions for the quasilinear Schrödinger system in RN in [9], which involves a parameter α and subcritical nonlinearities. By developing a new iterative technique and suitable estimation, the existence of the entire radial large solution was established for the modified quasilinear Schrödinger elliptic system in [10].

The study of System (1.1) was in part motivated by the nonlinear Schrödinger equation:

(1.3) iεz=εΔz+W(x)zl(z2)zkεΔh(z2)h(z2)z,xRN,

where W(x) is a given potential, k is a real constant, and l and h are real functions that are essentially pure power forms. The quasilinear Schrödinger equation (1.3) describes several physical phenomena with different h , see [1113] and references therein. We consider the case h(s)=s,l(s)=μsp12 , and k<0 . Setting z(t,x)=exp(iFt)u(x) , one obtains a corresponding equation of elliptic type which has the formal variational structure

(1.4) εΔu+V(x)uεk(Δu2)u=μup1u,u>0,xRN,

where V(x)=W(x)F is the new potential function. Problem (1.4) has caused a heated discussion. When k>0 is small enough, the existence result of multiple solutions was studied via dual approach techniques and variational methods [14]. The paper [15] established the existence of soliton solution by a minimization argument. The minimax principles for lower semicontinuous functionals, developed by Szulkin [16], were used to find solutions in [17]. The Mountain Pass Theorem combined with the principle of symmetric criticality to establish multiplicity of solutions in [18]. Moameni [19] changed variables to remove nonconvex term and created a suitable Orlicz space to meet the Mountain Pass Theorem, and proved the existence of soliton solution for a quasilinear Schrödinger equation involving critical exponent in RN .

Inspired by the above results, we apply the methods in [19] to solve system (1.1) and try to find nonnegative soliton solution. During this process, it is required that α,β are integers multiple of constant 2.

The main result of this article is the following:

Theorem 1.1

For system (1.1), k<0,N3 , α>2 , β>2 , α+β<2* , and α,β are integers multiple of constant 2, then there exists ε0>0 such that for all ε(0,ε0) , system (1.1) has a nonnegative solution (uε,vε)H1r and

(1.5) (uε(x),vε(x))(0,0),asε0.

The article is organized as follows. In Section 2, we reformulate this problem in an appropriate Orlicz space. In Section 3, we prove the existence of a solution for a special deformation of problem (1.1). Theorem 1.1 is proved in Section 4.

2 Reformulation of the problem and preliminaries

The energy functional associated with (1.1) is

Iε(u,v)=ε2RN(1ku2)u2dx+12RNu2dx+ε2RN(1kv2)v2dx+12RNv2dx2α+βRNuαvβdx.

By changing variables, we treated this problem in an Orlicz space. From [20] and [19], we changed variables as follows:

dz=k1ku2du,z=h(u)=12ku1ku2+12ln(ku+1ku2),dw=k1kv2dv,w=h(v)=12kv1kv2+12ln(kv+1kv2).

Since h is strictly monotone and has a well-defined inverse function u=f(z),v=f(w) . Note that

h(u){ku,u1k,k2uu,u1k,h(u)=k1ku2,h(v){kv,v1k,k2vv,v1k,h(v)=k1kv2,

and

f(z){1kz,z1z,2kzz,z1k,f(z)=1h(u)=1k1kv2=1k1kf(z)2,f(w){1kw,w1k,2kww,w1k,f(w)=1h(v)=1k1kv2=1k1kf(w)2.

Also, for some C0>0 it holds

G(t)=f(t)2{1kt2,t1k,2kt,t1k,G(2t)C0G(t),

G(t) is convex, G , i = 1 , 2 . Now we introduce the Orlicz space (see [21]):

E G ( R N ) = μ : R N G ( μ ) d x <

equipped with the norm:

μ G = inf ζ > 0 ζ 1 + R N G ( ζ 1 μ ( x ) ) d x .

Using this change of variable, we can rewrite the functional I ε ( u , v ) as

I ¯ ε ( z , w ) = ε 2 R N 1 k z 2 d x + 1 2 R N z 2 d x + ε 2 R N 1 k w 2 d x + 1 2 R N w 2 d x 2 α + β R N f ( z ) α f ( w ) β d x

defined in the space

H G 1 = ( z , w ) : R N z 2 d x < , R N G ( z ) d x < , R N w 2 d x < , R N G ( w ) d x < .

From the definition of G , only radially symmetric functions are in this space, and equipped with the norm:

( z , w ) = z L 2 + w L 2 + z G + w G ,

where L r is the Lebesgue function space with the norm

u L r = R N u r d x 1 r , 1 r < .

Here are some related facts:

Proposition 2.1

  1. E G ( R N ) is a Banach space.

  2. If μ n μ in E G ( R N ) , then R N G ( μ n ) G ( μ ) d x 0 and R N f ( μ n ) f ( μ ) 2 d x 0 .

  3. If μ n ( x ) μ ( x ) a.e. in R N and R N G ( μ n ) d x R N G ( μ ) d x , then μ n μ in E G ( R N ) .

  4. The dual space E G * ( R N ) = L L 2 = { ν : ν L , R N ν 2 d x < } .

  5. If μ E G ( R N ) , then ν = G ( μ ) = 2 f ( μ ) f ( μ ) , and ν E G * = sup ϕ G 1 ( ν , ϕ ) C 1 ( 1 + R N G ( μ ) d x ) , where C 1 is a constant independent of μ .

  6. For N > 2 , the map: ( μ 1 , μ 2 ) ( f ( μ 1 ) , f ( μ 2 ) ) from H G 1 into L q ( R N ) is continuous for 2 q 2 2 * and is compact for 2 < q < 2 2 * .

  7. Suppose B k is the ball with center at the coordinate origin and radius k > 0 . Let r < s and Q = B s \ B r . The map: ( μ 1 , μ 2 ) ( f ( μ 1 ) , f ( μ 2 ) ) from H G 1 into L q ( Q ) is compact for q 2 .

Proof

See [20] for the proof of parts (i)–(vi). The proof of part (vii) is similar to the proof of [19].□

Denote by H r 1 the space of radially symmetric functions in

H 1 , 2 = { u : u L 2 ( R N ) , u L 2 ( R N ) } .

Throughout this article, we use the standard notations. , H 1 , E G , L t , and stand for R N , H 1 , 2 , E G ( R N ) , L t ( R N ) , and H G 1 ( R N ) , respectively. We use C to denote any constant that is independent of the sequences considered, and the operation: ( a , b ) ( a , b ) = a a , b b , where ⁎ represents any operation.

3 Auxiliary problem

In this section, we show some results needed to prove Theorem 1.1. Indeed, we consider a special deformation H ¯ ε ( z , w ) (see (3.1) in the following) of I ¯ ε ( z , w ) first. The functional H ¯ ε ( z , w ) satisfies the PS condition and to which we can apply the Mountain Pass Theorem. Consequently, H ¯ ε ( z , w ) has a critical point for each ε > 0 . We can use it to prove Theorem 1.1 in the next section. In fact, we will see that the functionals I ¯ ε ( z , w ) and H ¯ ε ( z , w ) will coincide for the small values of ε . This idea was explored in [19,22].

To do this, we shall consider constants θ and l satisfying

4 < θ < 2 2 * , l > θ θ 2 .

Let a i > 0 be the value at which ξ i ( a i ) a i = 1 l , i = 1 , 2 , where ξ 1 = α u α 2 u v β , ξ 2 = β u α v β 2 v . Set

ξ i ¯ ( s ) = ξ i ( s ) , if s a i , 1 l s , if s > a i ,

and define

y i ( x , s i ) = χ Λ ξ i ( s i ) + ( 1 χ Λ ) ξ ¯ i ( s i ) ,

where χ Λ denotes the characteristic function of the set Λ , which is a bounded domain. For convenience of later calculation, let Λ be a ring domain. Set Y i ( x , t i ) = 0 t i y i ( x , ς ) d ς , where i = 1 , 2 . Set y ( x , s 1 , s 2 ) = y 1 ( x , s 1 ) + y 2 ( x , s 2 ) , Y ( x , s 1 , s 2 ) = Y 1 ( x , s 1 ) + Y 2 ( x , s 2 ) . According to [19], the functions y , Y satisfy the following conditions:

0 θ Y ( x , s 1 , s 2 ) y ( x , s 1 , s 2 ) ( s 1 , s 2 ) , x Λ , s i 0 , i = 1 , 2 , ( y 1 )

0 2 Y ( x , s 1 , s 2 ) y ( x , s 1 , s 2 ) ( s 1 , s 2 ) 1 l ( s 1 2 , s 2 2 ) , x Λ c , s i R , i = 1 , 2 . ( y 2 )

Using the modification, the energy functional I ε ( u , v ) becomes

H ε ( u , v ) = ε 2 ( 1 k u 2 ) u 2 d x + 1 2 u 2 d x + ε 2 ( 1 k v 2 ) v 2 d x + 1 2 v 2 d x 2 α + β Y ( x , u , v ) d x .

As in Section 2, we can rewrite the functional H ε ( u , v ) as a new functional H ¯ ε ( z , w ) as follows:

(3.1) H ¯ ε ( z , w ) = ε 2 1 k z 2 d x + 1 2 f ( z ) 2 d x + ε 2 1 k w 2 d x + 1 2 f ( w ) 2 d x 2 α + β Y ( x , f ( z ) , f ( w ) ) d x .

H ¯ ε ( z , w ) is defined on the Orlicz space H G 1 . In this section, we shall assume ε = 1 , H 1 = H , H ¯ 1 = H ¯ .

Some properties of the functional H ¯ are stated as follows:

Proposition 3.1

  1. H ¯ is well defined in H G 1 .

  2. H ¯ is continuous in H G 1 .

  3. H ¯ is Gateaux-differentiable in H G 1 .

Proof

The proof is similar to the proof of [20].□

The following theorem is the main result in this section.

Theorem 3.1

H ¯ has a critical point in H G 1 , that is, there exists ( 0 , 0 ) ( z , w ) H G 1 such that

(3.2) 1 k z ϕ d x + f ( z ) f ( z ) ϕ d x + 1 k w ψ d x + f ( w ) f ( w ) ψ d x 2 α + β y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) ( ϕ , ψ ) d x = 0 ,

for every ( ϕ , ψ ) H G 1 .

We will use the Mountain Pass Theorem like [23,24]. First, let us define the Mountain Pass value

C 0 inf γ τ sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ¯ ( γ ( t 1 , t 2 ) ) ,

where

τ = { γ C ( [ 0 , 1 ] × [ 0 , 1 ] , H G 1 ) : γ ( 0 , 0 ) = 0 , H ¯ ( γ ( 1 , 1 ) ) 0 , γ ( 1 , 1 ) 0 } .

Next, we will prove Theorem 3.1 by the following lemmas.

Lemma 3.1

The functional H ¯ satisfies the Mountain Pass geometry.

Proof

From the definition of Y , we have H ¯ ( 0 , 0 ) = 0 . Clearly, there exists ( z 0 , w 0 ) that satisfies H ¯ ( z 0 , w 0 ) > 0 . Let e 1 , e 2 C 0 , r ( R N ) with e 0 and supp ( e i ) Ω , i = 1 , 2 . It is easy to see that H ( t e 1 , t e 2 ) 0 for the large values of t . Consequently, there exists ( 0 , 0 ) ( z , w ) H G 1 such that H ¯ ( z , w ) 0 , where z = h ( t e 1 ) , w = h ( t e 2 ) . Therefore, the functional H ¯ satisfies the Mountain Pass Geometry.□

This lemma guaranties the existence of a ( PS ) C 0 sequence ( z n , w n ) , that is, H ¯ ( z n , w n ) C 0 and H ¯ ( z n , w n ) 0 .

Lemma 3.2

C 0 is positive.

Proof

Set

S ρ ( z , w ) H G 1 : ε 1 k z 2 d x + f ( z ) 2 d x = ρ 2 , ε 1 k w 2 d x + f ( w ) 2 d x = ρ 2 ,

where 0 < ρ 1 .

For ( z , w ) S ρ , we have f ( z ) 2 d x ρ 2 , f ( w ) 2 d x ρ 2 . By Hölder inequality and continuity of f , we obtain

(3.3) f ( z ) α d x f ( z ) 2 d x α 2 ρ α ,

(3.4) f ( w ) β d x f ( w ) 2 d x β 2 ρ β .

Also, it follows from ( y 1 ) and ( y 2 ) that

(3.5) Y ( x , f ( z ) , f ( w ) ) d x = Λ Y ( x , f ( z ) , f ( w ) ) d x + Λ c Y ( x , f ( z ) , f ( w ) ) d x 1 θ f ( z ) α d x + f ( w ) β d x + 1 2 l f ( z ) 2 d x + f ( w ) 2 d x .

From (3.3), (3.4), and (3.5), we obtain

H ¯ ε ( z , w ) = ε 2 1 k z 2 d x + 1 2 f ( z ) 2 d x + ε 2 1 k w 2 d x + 1 2 f ( w ) 2 d x 2 α + β Y ( x , f ( z ) , f ( w ) ) d x ρ 2 2 ( α + β ) θ ( ρ α + ρ β ) 2 ( α + β ) l ρ 2 1 2 ( α + β ) l ρ 2 2 ( α + β ) θ ( ρ α + ρ β ) C ¯ ρ 2 ,

if 0 < ρ ρ 0 1 for some ρ 0 , then C ¯ = ( 1 2 ( α + β ) l 2 ( α + β ) θ ) > 0 . Hence, for ( z , w ) S ρ with 0 < ρ ρ 0 1 we have

(3.6) H ¯ ε ( z , w ) C ¯ ρ 2 .

If γ ( 1 ) = ( z , w ) and H ¯ ( γ ( 1 ) ) < 0 , then it follows from (3.6) that

1 k z 2 d x + f ( z ) 2 d x > ρ 0 2 , 1 k w 2 d x + f ( w ) 2 d x > ρ 0 2 ,

thereby giving

sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ¯ ( γ ( t 1 , t 2 ) ) sup γ ( t 1 , t 2 ) S ρ H ¯ ( γ ( t 1 , t 2 ) ) C ¯ ρ 0 2 ,

which combines the definition of C 0 , and we have

C 0 C ¯ ρ 0 2 > 0 .

Lemma 3.3

Suppose ( z n , w n ) is a ( PS ) C 0 sequence. The following statements hold:

  1. ( z n , w n ) is bounded in H G 1 .

  2. For each δ > 0 , there exists R > 0 , such that

    limsup n + B R c ε k z n 2 + f ( z n ) 2 d x < δ ,

    limsup n + B R c ε k w n 2 + f ( w n ) 2 d x < δ .

  3. If ( z n , w n ) converges weakly to ( z , w ) in H G 1 , then

    lim n + y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x = y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x .

  4. If ( z n , w n ) 0 converges weakly to ( z , w ) in H G 1 , then for every nonnegative test function ( ϕ , ψ ) H G 1 we have

    lim n + H ¯ ( z n , w n ) , ( ϕ , ψ ) = H ¯ ( z , w ) , ( ϕ , ψ ) .

Proof

Since ( z n , w n ) is a ( PS ) C 0 sequence, we have

(3.7) H ¯ ε ( z n , w n ) = ε 2 1 k z n 2 d x + 1 2 f ( z n ) 2 d x + ε 2 1 k w n 2 d x + 1 2 f ( w n ) 2 d x 2 α + β Y ( x , f ( z n ) , f ( w n ) ) d x = C 0 + o ( 1 )

and

(3.8) H ¯ ε ( z n , w n ) , ( ϕ , ψ ) = ε 1 k z ϕ d x + f ( z n ) f ( z n ) ϕ d x + ε 1 k w n ψ d x + f ( w n ) f ( w n ) ψ d x 2 α + β y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) d x = o ( ( ϕ , ψ ) ) .

For part (i), pick

( ϕ , ψ ) = f ( z n ) f ( z n ) , f ( w n ) f ( w n ) = ( k 1 f 2 ( z n ) f ( z n ) , k 1 f 2 ( w n ) f ( w n ) ) .

It is easy to see that ( ϕ , ψ ) G C ( z n , w n ) G and

( ϕ , ψ ) = 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n , 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n ( 2 z n , 2 w n ) ,

hence, ( ϕ , ψ ) C ( z n , w n ) . Substituting ( ϕ , ψ ) in (3.8), then

(3.9) H ¯ ε ( z n , w n ) , f ( z n ) f ( z n ) , f ( w n ) f ( w n ) = ε 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 d x + f ( z n ) 2 d x + ε 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 d x + f ( w n ) 2 d x 2 α + β y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x = o ( ( f ( z n ) , f ( w n ) ) ) .

It follows from ( y 1 ) and ( y 2 ) that

(3.10) G ( x , f ( z n ) , f ( w n ) ) d x + 1 θ y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) d x 1 l 1 θ 1 2 ( f ( z n ) 2 + f ( w n ) 2 ) d x .

Taking into account (3.7), (3.9), and (3.10), we obtain

C 0 + o ( 1 ) + o ( ( ϕ , ψ ) ) = H ¯ ε ( z n , w n ) 1 θ H ¯ ε ( z n , w n ) , f ( z n ) f ( z n ) , f ( w n ) f ( w n ) ε 2 2 ε θ 1 k z n 2 d x + 1 2 1 θ f ( z n ) 2 d x + ε 2 2 ε θ 1 k w n 2 d x + 1 2 1 θ f ( w n ) 2 d x + 2 α + β 1 θ y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) Y ( x , f ( z n ) , f ( w n ) ) d x ε 2 2 ε θ 1 k z n 2 d x + ε 2 2 ε θ 1 k w n 2 d x + 1 1 l 1 θ 1 2 ( f ( z n ) 2 + f ( w n ) 2 ) d x .

Since ε 2 2 ε θ > 0 and ( 1 1 l ) ( 1 θ 1 2 ) > 0 , it follows from the above that z n 2 d x + f ( z n ) 2 d x + w n 2 d x + f ( w n ) 2 d x is bounded. It proves part (i).

For part (ii), let η R C ( R N ) be a function satisfying η R = 0 on B R 2 , η R = 1 on B R c , and η R ( x ) C R . It follows from part (i) that ( z n , w n ) is bounded in H G 1 . Hence, from (3.8) we have

H ¯ ε ( z n , w n ) , f ( z n ) f ( z n ) η R , f ( w n ) f ( w n ) η R = o ( 1 ) ,

which stands for

ε 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 η R d x + f ( z n ) 2 η R d x + ε k f ( z n ) f ( z n ) z n η R d x + ε 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 η R d x + f ( w n ) 2 η R d x + ε k f ( w n ) f ( w n ) w n η R d x = 2 α + β y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) η R , f ( w n ) η R ) d x .

By ( y 2 ) , we obtain

y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) 1 l ( f 2 ( z n ) , f 2 ( w n ) ) , x B R 2 c .

Therefore,

(3.11) ε 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 η R d x + ε 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 η R d x + 1 1 l 2 α + β ( f ( z n ) 2 η R , f ( w n ) 2 η R ) d x ε k C R f ( z n ) f ( z n ) z n , f ( w n ) f ( w n ) w n d x + o ( 1 ) ε k C R ( z n , w n ) d x + ε k C R ( f ( z n ) 2 + f ( z n ) 4 , f ( w n ) 2 + f ( w n ) 4 ) d x + o ( 1 ) .

Also, like [19], it follows from Proposition 2.1(vi) that { f ( v n ) } n is a bounded sequence in L 2 L 2 2 * . Hence, ( f ( z n ) 2 + f ( z n ) 4 , f ( w n ) 2 + f ( w n ) 4 ) d x is bounded. Therefore, it follows from (3.11) that

limsup n + B R c ε k z n 2 + f ( z n ) 2 d x < δ ,

limsup n + B R c ε k w n 2 + f ( w n ) 2 d x < δ .

It proves part (ii).

For part (iii), from part (ii) we know that for each δ > 0 , there exists R > 0 , such that

(3.12) limsup n + B R c ε k z n 2 + f ( z n ) 2 d x < l δ 4 ,

(3.13) limsup n + B R c ε k w n 2 + f ( w n ) 2 d x < l δ 4 .

We might as well make B R c Λ c , R 1 < R , B R 1 Λ c , and it follows from ( y 2 ) that

y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) 1 l ( f ( z n ) 2 , f ( w n ) 2 ) , x B R c ,

and from (3.12), (3.13), we obtain

(3.14) limsup n + B R c y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x δ 4 , δ 4

and

(3.15) B R c y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x δ 4 , δ 4 .

By (3.14) and (3.15), we have

(3.16) y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x δ 2 + B R 1 y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x + B R \ B R 1 y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x .

Because of B R 1 Λ c ,

y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) 1 l ( f ( z n 2 ) , f ( w n ) 2 ) , x B R 1 .

Then, by the compact embedding theorem and Lebesgue theorem, we obtain a subsequence still denoted by ( f ( z n ) , f ( w n ) ) , such that

(3.17) B R 1 y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x B R 1 y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x .

It follows from part (vii) of Proposition 2.1 that the map: ( μ 1 , μ 2 ) ( f ( μ 1 ) , f ( μ 2 ) ) from H G 1 into L q ( B R \ B R 1 ) is compact for q 2 , combined with ( y 2 ) , hence

(3.18) B R \ B ¯ R 1 y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x B R \ B ¯ R 1 y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x .

Considering (3.17) and (3.18), it follows from (3.16) that

limsup n + y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x δ 2

for every δ > 0 . Consequently,

y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x

as n . It proves part (iii).

For part (iv), we know f is increasing and f = 0 , hence f ( z n ) 0 , f ( w n ) 0 and f ( z ) 0 , f ( w ) 0 . For the second term and the fourth term on the right-hand side of (3.8), we have

f ( z n ) f ( z n ) ϕ f ( z n ) ϕ ,

f ( w n ) f ( w n ) ψ f ( w n ) ψ ,

and since ( z n , w n ) ( z , w ) weakly in H G 1 , for the right-hand side of the above two inequalities we have

lim n + f ( z n ) ϕ d x = f ( z ) ϕ d x ,

lim n + f ( w n ) ψ d x = f ( w ) ψ d x .

By the dominated convergence theorem and the fact that ( z n , w n ) ( z , w ) a.e. in R N , we obtain

(3.19) lim n + f ( z n ) f ( z n ) ϕ d x = f ( z ) f ( z ) ϕ d x

and

(3.20) lim n + f ( w n ) f ( w n ) ψ d x = f ( w ) f ( w ) ψ d x .

For the fifth term on the right-hand side of (3.8), we have

y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) 1 l ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) , x Λ c ,

and similarly by the dominated convergence theorem, we obtain

(3.21) lim n + Λ c y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) d x = Λ c y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) ( ϕ , ψ ) d x .

Also, we know

y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) ( u α 2 v β , u α v β 2 ) ( ϕ , ψ ) , x Λ

and follows from part (vii) of Proposition 2.1 that the map: ( μ 1 , μ 2 ) ( f ( μ 1 ) , f ( μ 2 ) ) from H G 1 into L q ( Λ ) is compact for q 2 , hence it follows from the dominated convergence theorem that

(3.22) lim n + Λ y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) ( ϕ , ψ ) d x = Λ y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) ( ϕ , ψ ) d x .

It follows from (3.8) and (3.24)–(3.27) that

lim n + H ¯ ε ( z n , w n ) , ( ϕ , ψ ) = H ¯ ε ( z , w ) , ( ϕ , ψ ) .

It proves part (iv).□

Lemma 3.4

If ( z n , w n ) is a ( PS ) C 0 sequence, then ( z n , w n ) 0 converges to ( z , w ) in H G 1 . Consequently, H ¯ ( z , w ) = lim n + H ¯ ( z n , w n ) and H ¯ ( z , w ) = 0 .

Proof

It follows from part (i) of Lemma 3.3 that ( z n , w n ) is bounded in H G 1 . Hence, there exists ( z , w ) H G 1 such that, up to a subsequence, ( z n , w n ) ( z , w ) weakly in H G 1 and ( z n , w n ) ( z , w ) a.e. in R N . We may replace ( z n , w n ) by ( z n , w n ) , hence ( z n , w n ) 0 and ( z , w ) 0 . Since ( z n , w n ) is a ( PS ) C 0 sequence, we have

(3.23) o ( ( f 1 ( z n ) , f 2 ( w n ) ) ) = H ¯ ε ( z n , w n ) , f ( z n ) f ( z n ) , f ( w n ) f ( w n ) = ε 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 d x + f ( z n ) 2 d x + ε 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 d x + f ( w n ) 2 d x 2 α + β y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x

and

(3.24) o ( ( f ( z ) , f ( w ) ) ) = H ¯ ε ( z n , w n ) , f ( z ) f ( z ) , f ( w ) f ( w ) .

It follows from part (iv) of Lemma 3.3 and (3.24) that

(3.25) H ¯ ε ( z n , w n ) , f ( z ) f ( z ) , f ( w ) f ( w ) = H ¯ ε ( z , w ) , f ( z ) f ( z ) , f ( w ) f ( w ) + o ( ( f ( z ) , f ( w ) ) ) = ε 1 k 1 + k f 2 ( z ) 1 k f 2 ( z ) z 2 d x + f ( z ) 2 d x + ε 1 k 1 + k f 2 ( w ) 1 k f 2 ( w ) w 2 d x + f ( w ) 2 d x 2 α + β y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x + o ( ( f ( z ) , f ( w ) ) ) .

According to [19], we also have

(3.26) ε 1 k k f 2 ( z ) z 2 1 k f 2 ( z ) d x liminf n + ε 1 k k f 2 ( z n ) z n 2 1 k f 2 ( z n ) d x ,

(3.27) ε 1 k k f 2 ( w ) w 2 1 k f 2 ( w ) d x liminf n + ε 1 k k f 2 ( w n ) w n 2 1 k f 2 ( w n ) d x .

Also, lower semi continuity and Fatou lemma imply

(3.28) ε z 2 d x liminf n + ε z n 2 d x ,

(3.29) ε w 2 d x liminf n + ε w n 2 d x ,

(3.30) G ( z ) 2 d x = liminf n + G ( z n ) 2 d x ,

(3.31) G ( w ) 2 d x = liminf n + G ( w n ) 2 d x .

Up to a subsequence one can assume

(3.32) liminf n + ε z n 2 d x = lim n + ε z n 2 d x ,

(3.33) liminf n + G ( z n ) 2 d x = lim n + G ( z n ) 2 d x ,

(3.34) liminf n + ε 1 k k f 2 ( z n ) z n 2 1 k f 2 ( z n ) d x = lim n + ε 1 k k f 2 ( z n ) z n 2 1 k f 2 ( z n ) d x .

There exist nonnegative numbers δ 1 , δ 2 , and δ 3 such that

(3.35) lim n + ε z n 2 d x = ε z 2 d x + δ 1 ,

(3.36) lim n + G ( z n ) 2 d x = G ( z ) 2 d x + δ 2 ,

(3.37) lim n + ε 1 k k f 2 ( z n ) z n 2 1 k f 2 ( z n ) d x = ε 1 k k f 2 ( z ) z 2 1 k f 2 ( z ) d x + δ 3 .

Continuing in this way, we can obtain

(3.38) lim n + ε w n 2 d x = ε w 2 d x + δ 1 ,

(3.39) lim n + G ( w n ) 2 d x = G ( w ) 2 d x + δ 2 ,

(3.40) lim n + ε 1 k k f 2 ( w n ) w n 2 1 k f 2 ( w n ) d x = ε 1 k k f 2 ( w ) w 2 1 k f 2 ( w ) d x + δ 3 ,

where δ 1 , δ 2 , and δ 3 are the nonnegative numbers.

It follows from part (iii) of Lemma 3.3 that

lim n + y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x = y ( x , f ( z ) , f ( w ) ) ( f ( z ) , f ( w ) ) d x ,

which together with (3.23) and (3.25) imply

(3.41) lim n + ε 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 d x + f ( z n ) 2 d x + ε 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 d x + f ( w n ) 2 d x = lim n + 2 α + β y ( x , f ( z n ) , f ( w n ) ) ( f ( z n ) , f ( w n ) ) d x .

Taking into account (3.35)–(3.40), the above limit implies δ 1 = δ 2 = δ 3 = δ 1 = δ 2 = δ 3 = 0 . Therefore, it follows from (3.35), (3.36), (3.38), and (3.39) that

lim n + ε z n 2 d x = ε z 2 d x , lim n + G ( z n ) 2 d x = G ( z ) 2 d x , lim n + ε w n 2 d x = ε w 2 d x , lim n + G ( w n ) 2 d x = G ( w ) 2 d x .

By Proposition 2.1, ( z n , w n ) ( z , w ) in E G and we have ( z n , w n ) ( z , w ) in L 2 × L 2 . Hence, ( z n , w n ) ( z , w ) in H G 1 .□

By Lemmas 3.1, 3.2, and 3.4, we obtain Theorem 3.1.

4 Proof of Theorem 1.1

To prove Theorem 1.1, we need to find a critical point for the functional I ¯ ε . It follows from [19] that the functionals I ¯ ε and H ¯ ε will coincide for the small values of ε , every critical point of H ¯ ε will be a critical point of I ¯ ε .

Without loss of generality, assume ε 2 instead of ε in the functionals H ¯ ε and I ¯ ε , i.e.,

H ¯ ε ( z , w ) = ε 2 2 1 k z 2 d x + 1 2 f ( z ) 2 d x + ε 2 2 1 k w 2 d x + 1 2 f ( w ) 2 d x 2 α + β Y ( x , f ( z ) , f ( w ) ) d x ,

and

I ¯ ε ( z , w ) = ε 2 2 1 k z 2 d x + 1 2 z 2 d x + ε 2 2 1 k w 2 d x + 1 2 w 2 d x 2 α + β f ( z ) α f ( w ) β d x .

It follows from Theorem 3.1 that there exists a critical point ( z ε , w ε ) H G 1 of H ¯ ε for each ε > 0 . Set ( u ε , v ε ) = ( f ( z ε ) , f ( w ε ) ) .

The following lemmas are crucial for the proof of Theorem 1.1.

Lemma 4.1

The sequence ( u ε , v ε ) ε > 0 is strongly convergent to ( 0 , 0 ) when ε 0 , in H 1 × H 1 , i.e.,

( u ε , v ε ) H 1 × H 1 0 , as ε 0 .

Proof

Let ( 0 , 0 ) ( ϕ , ψ ) C 0 , r be a nonnegative function with ( supp ( ϕ ) , supp ( ψ ) ) Ω , and H 1 ( ( ϕ , ψ ) ) 0 . Set γ 1 ( t 1 , t 2 ) h ( t 1 ϕ , t 2 ψ ) . Hence, we have

H ¯ ε ( γ 1 ( 1 , 1 ) ) = H ¯ ε ( h ( ϕ , ψ ) ) = H ε ( ϕ , ψ ) H 1 ( ϕ , ψ ) 0 .

It follows from the definition of the Mountain Pass value that

H ¯ ε ( z ε , w ε ) = inf γ τ sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ¯ ε ( γ ( t 1 , t 2 ) ) sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ¯ ε ( γ 1 ( t 1 , t 2 ) ) = sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ¯ ε ( h ( t 1 ϕ , t 2 ψ ) ) = sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ε ( t 1 ϕ , t 2 ψ ) .

And following the maximum property of function, we obtain

(4.1) H ¯ ε ( z ε , w ε ) sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] H ε ( t 1 ϕ , t 2 ψ ) = sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] ε 2 t 1 4 2 ( k ϕ 2 ) ϕ 2 d x + ε 2 t 1 2 2 ϕ 2 d x + t 1 2 2 ϕ 2 d x + ε 2 t 2 4 2 ( k ψ 2 ) ψ 2 d x + ε 2 t 2 2 2 ψ 2 d x + t 2 2 2 ψ 2 d x 2 ( t 1 α t 2 β ) α + β ϕ α ψ β d x sup ( t 1 , t 2 ) [ 0 , 1 ] × [ 0 , 1 ] ε 2 t 1 2 2 ( 1 k ϕ 2 ) ϕ 2 d x + t 1 2 2 ϕ 2 d x + ε 2 t 2 2 2 ( 1 k ψ 2 ) ψ 2 d x + t 2 2 2 ψ 2 d x 2 t 1 α t 2 β α + β ϕ α ψ β d x ε C A ( ϕ , ψ ) ,

where A ( ϕ , ψ ) is a function about ( ϕ , ψ ) , C > 4 . Now, as in the proof of part (i) of Lemma 3.3 we obtain

(4.2) H ¯ ε ( z n , w n ) = H ¯ ε ( z n , w n ) 1 θ H ¯ ε ( z n , w n ) , f ( z n ) f ( z n ) , f ( w n ) f ( w n ) ε 2 1 2 2 θ 1 k z n 2 d x + ε 2 1 2 2 θ 1 k w n 2 d x + 1 1 l 1 θ 1 2 ( f ( z n ) 2 + f ( w n ) 2 ) d x .

Combining (4.1) and (4.3), we obtain

ε 2 1 2 2 θ 1 k z n 2 d x + ε 2 1 2 2 θ 1 k w n 2 d x + 1 1 l 1 θ 1 2 ( f ( z n ) 2 + f ( w n ) 2 ) d x ε C A ( ϕ , ψ ) .

Therefore,

(4.3) 1 2 2 θ 1 k z n 2 d x + ε 2 1 2 2 θ 1 k w n 2 d x + 1 1 l 1 θ 1 2 ( f ( z n ) 2 + f ( w n ) 2 ) d x ε C 2 A ( ϕ , ψ ) .

Hence, substituting ( u ε , v ε ) = ( f ( z ε ) , f ( w ε ) ) in (4.3), then

(4.4) ( 1 + u ε 2 ) u ε 2 d x + ( 1 + v ε 2 ) v ε 2 d x + ( u ε 2 + v ε 2 ) d x C ε C 2 A ( ϕ , ψ ) .

Therefore,

( u ε , v ε ) H 1 × H 1 0 , as ε 0 .

Lemma 4.2

Let N > 2 . There is a constant C = C N , such that

0 u ε ( x ) C x N 2 N u ε ( x ) H 1 , x 0 , 0 v ε ( x ) C x N 2 N v ε ( x ) H 1 , x 0 ,

for any ( u , v ) H r 1 × H r 1 .

This lemma is liking [25].

Lemma 4.3

For every compact set Q R N such that Q is nonempty, ( u ε , v ε ) L ( Q ) × L ( Q ) 0 as ε 0 .

Proof

For each ε > 0 , it follows from Lemma 4.2 that

0 u ε ( x ) C x N 2 N u ε ( x ) H 1 , x 0 ,

0 v ε ( x ) C x N 2 N v ε ( x ) H 1 , x 0 ,

which together with the result of Lemma 4.1 obviously means

( u ε , v ε ) L ( Q ) × L ( Q ) 0 , as ε 0 .

Next, we prove that Theorem 1.1.

Proof

By Lemma 4.3, we have

(4.5) max x Λ ( f ( z ε ) , f ( w ε ) ) ( 0 , 0 ) , as ε 0 .

From (4.5), N 1 , N 2 > 0 , there exists ε 0 > 0 , such that max x Λ ( f ( z ε ) , f ( w ε ) ) < ( N 1 , N 2 ) for every 0 < ε < ε 0 . Using the test function ( ϕ , ψ ) = ( f ( z ε ) N 1 ) + f ( z ε ) , ( f ( w ε ) N 2 ) + f ( w ε ) , we obtain

0 = H ¯ ε ( z n , w n ) , ( f ( z ε ) N ) + f ( z ε ) , ( f ( w ε ) N ) + f ( w ε ) = ε 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 d x + f ( z n ) ( f ( z ε ) N ) + d x + ε 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 d x + f ( w n ) ( f ( w ε ) N ) + d x 2 α + β y ( x , f ( z n ) , f ( w n ) ) ( ( f ( z ε ) N ) + , ( f ( w ε ) N ) + ) d x

= ε F 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 d x + R N \ Λ ¯ f ( z n ) ( f ( z ε ) N ) + d x + ε F 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 d x + R N \ Λ ¯ f ( w n ) ( f ( w ε ) N ) + d x 2 α + β R N \ Λ ¯ y ( x , f ( z n ) , f ( w n ) ) ( ( f ( z ε ) N ) + , ( f ( w ε ) N ) + ) d x ,

where F = ( R N \ Λ ¯ ) { x : ( f ( z ε ) , f ( w ε ) ) ( N 1 , N 2 ) } . From ( y 2 ) , we have

( f ( z n ) , f ( w n ) ) ( ( f ( z ε ) N ) + , ( f ( w ε ) N ) + ) y ( x , f ( z n ) , f ( w n ) ) ( ( f ( z ε ) N ) + , ( f ( w ε ) N ) + ) 0 , x Λ c .

Thus,

ε F 1 k 1 + k f 2 ( z n ) 1 k f 2 ( z n ) z n 2 d x + ε F 1 k 1 + k f 2 ( w n ) 1 k f 2 ( w n ) w n 2 d x = 0 ,

from which we obtain

( f ( z ε ) , f ( w ε ) ) ( N 1 , N 2 ) , x R N \ Λ ¯ .

Let ( N 1 , N 2 ) = ( a 1 , a 2 ) , therefore

y ( x , f ( z n ) , f ( w n ) ) = ( α f α 1 ( z n ) f β ( w n ) , β f α ( z n ) f β 1 ( w n ) ) , x R N \ Λ ¯ ,

and it follows that H ¯ ε ( z n , w n ) , ( ϕ , ψ ) = 0 , we have

ε 2 1 k z ε ϕ d x + f ( z ε ) f ( z ε ) ϕ d x + ε 2 1 k w ε ψ d x + f ( w ε ) f ( w ε ) ψ d x = 2 α + β ( α f ( z n ) α 2 f ( z n ) f ( w n ) β ϕ , β f ( z n ) α f ( w n ) β 2 f ( w n ) ψ ) ( f ( z n ) , f ( w n ) ) d x ,

for every ( ϕ , ψ ) H G 1 and 0 < ε < ε 0 . Therefore, I ¯ ε ( z , w ) has a critical point ( z ε , w ε ) in H G 1 for every ε ( 0 , ε 0 ) .□

  1. Funding information: Jing Zhang was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (No. 2022MS01001, 2023LHMS01005), Key Laboratory of Infinite-dimensional Hamiltonian System and Its Algorithm Application (Inner Mongolia Normal University), Ministry of Education (No. 2023KFZD01), Research Program of science and technology at Universities of Inner Mongolia Autonomous Region (No. NJYT23100), Mathematics First-class Disciplines Cultivation Fund of Inner Mongolia Normal University (No. 2024YLKY14), and the Fundamental Research Funds for the Inner Mongolia Normal University (No. 2022JBQN072). Xue Zhang was supported by the Fundamental Research Funds for the Inner Mongolia Normal University (2022JBXC03) and Graduate students’ research innovation fund of Inner Mongolia Normal University (CXJJS22100).

  2. Author contributions: The authors declare that they have equal contributions.

  3. Conflict of interest: The authors state no conflicts of interest.

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Received: 2023-11-07
Revised: 2024-04-28
Accepted: 2024-05-16
Published Online: 2024-07-22

© 2024 the author(s), published by De Gruyter

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

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  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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