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Classification of positive solutions for a weighted integral system on the half-space

  • Qiuping Liao , Haofeng Wang and Yingying Xiao EMAIL logo
Published/Copyright: September 16, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we study the following weighted integral system:

u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 , v ( x ) = ∫ R + n + 1 y n + 1 β g ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 .

Under nature structure conditions on f and g , we classify the positive solutions using the method of moving spheres.

Keywords: method of moving spheres; weighted integral system
MSC 2010: 45G15; 45M20

1 Introduction

The classical Hardy-Littlewood-Sobolev (HLS) inequality, as stated in [1,2], is given by

(1) ∫ R + n ∫ R + n f ( x ) ∣ x − y ∣ − λ g ( y ) d x d y ≤ C α , n , p ∥ f ∥ L p ( R n ) ∥ g ∥ L q ′ ( R n ) ,

for all f ∈ L p ( R n ) , g ∈ L q ′ ( R n ) , 1 < p , q ′ < ∞ , 1 p + 1 p ′ = 1 , 1 q + 1 q ′ = 1 , 0 < λ < n , and 1 p + 1 q ′ + λ n = 2 . This inequality mentioned has become an indispensable tool in the study of partial differential equations. The categorization of extremal functions plays a significant role in determining the best constant. Lieb utilized the rearrangement inequality to prove the existence of the optimal function for inequality (1) in his publication [3]. Moreover, he provided a classification of the best functions and computed the sharp constant in cases where either p or q ′ is equal to 2 or p = q ′ .

Inequality (1) has profound implications in the domain of geometric inequalities, encompassing noteworthy results such as the Gross logarithmic Sobolev inequality and the Moser-Onofri-Beckner inequality, as elucidated by Beckner [4]. Another fundamental finding pertains to the equivalence between the HLS inequality and the Sobolev inequality, which can be established by selecting λ = n − 2 , p = q ′ = 2 n n + 2 , and utilizing Green’s representation formula. Extensive advancements have been made in extending these inequalities to diverse contexts. Remarkable contributions have emerged from researchers such as Jerison and Lee [5], Frank and Lieb [6], and Han et al. [7] concerning extensions on Heisenberg groups. Dou and Zhu [8] have explored extensions on upper half spaces, while Han and Zhu [9] have attained results on compact Riemannian manifolds. Moreover, several works, including [10–13], have established the reversed (weighted) HLS inequality. For a more comprehensive understanding of the weighted HLS inequality and Hardy-Sobolev equations, it is recommended to consult the following references: [5–7,14–23]. These sources provide additional insights and advancements in this area of study.

In the work of Gluck [24], a significant breakthrough was made in establishing the precise HLS inequality specifically for the half-space, taking into account a general kernel and conformal invariance. This remarkable result holds true when the parameters p and q ′ satisfy the conditions p = 2 ( n − 1 ) 2 ( n − 1 ) − λ and q ′ = 2 n 2 n − λ + 2 γ .

(2) ∫ R + n ∫ R + n K λ , γ ( x ′ − y , x n ) f ( y ) g ( x ) d x d y ≤ C n , λ , γ , p ∥ f ∥ L p ( ∂ R + n ) ∥ g ∥ L q ′ ( R + n ) ,

where the kernel

K λ , γ ( x ′ , x n ) = x n γ ( ∣ x ′ ∣ 2 + x n 2 ) λ 2 , x = ( x ′ , x n ) ∈ R n − 1 × ( 0 , ∞ ) ,

with γ ≥ 0 , 0 < n − λ + γ < n − γ ,

λ − 2 γ 2 n + λ 2 ( n − 1 ) < 1

and

R + n = { x = ( x 1 , … , x n ) ∈ R n ∣ x n > 0 } .

He classified all extremal functions of inequality (2) and then computed the best constant.

The family of kernels K λ , γ where λ > 0 , includes several well-known examples such as the classical Poisson kernel K n , 1 , the Riesz kernel K λ , γ , and the Poisson kernel K λ , 1 − n + λ for the divergence form operator u ↦ div ( x n n − λ ∇ u ) on the half-space. In [8,25–27], the corresponding inequalities to (2) for the kernels K n , 1 , K λ , 0 , K n − λ , 1 − n + λ , and K n − λ , 1 were investigated. In those works, the authors established the existence of extremal functions and subsequently classified all such extremal functions.

Hu and Liu [28] investigated the classification of positive solutions for the following integral system:

u ( y ) = ∫ R + n x n γ f ( v ( x ) ) ∣ x − y ∣ λ d x , y ∈ ∂ R + n , v ( x ) = ∫ ∂ R + n x n γ g ( u ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n ,

where n ≥ 2 , γ ≥ 0 , 0 < n − λ + γ < n − γ , λ − 2 γ 2 n + λ 2 ( n − 1 ) < 1 .

In a closely related context to the Stein-Weiss inequality [23], Dou and Ma [29] derived the following inequality:

(3) ∫ R + n + 1 ∫ R + n + 1 x n + 1 α y n + 1 β f ( y ) g ( x ) ∣ x − y ∣ λ d y d x ≤ C n , α , β , p ∥ f ∥ L p ( R + n + 1 ) ∥ g ∥ L q ′ ( R + n + 1 ) ,

where f ∈ L p ( R + n + 1 ¯ ) , g ∈ L q ′ ( R + n + 1 ¯ ) and λ , α , β , p , and q ′ satisfy

0 < λ < n + 1 , 1 < q ′ , p < ∞ , β > − 1 p ′ , α > − 1 q , α + β ≤ 0 , 1 p + 1 q ′ + λ − α − β n + 1 = 2 .

By employing the concentration-compactness principle, researchers successfully established the existence of extremal functions in their analysis. In the specific scenario of conformal invariance, they utilized the method of moving spheres to classify all such extremal functions. Recently, a study conducted by Dou et al. [30] introduced a converse form of inequality (3), thereby providing a new insight into the problem. Through the innovative implementation of the renormalization method, they not only demonstrated the existence of extremal functions but also achieved a comprehensive classification of all extremal functions. This breakthrough brings forth valuable contributions to the understanding of the problem at hand.

Hu and Du [31] studied the classification of positive solutions for following integral system:

u ( x ) = ∫ R + n + 1 x n + 1 α y n + 1 β f ( v ( y ) ) ∣ x − y ∣ λ d y , y ∈ R + n + 1 , v ( y ) = ∫ R + n + 1 x n + 1 α y n + 1 β g ( u ( x ) ) ∣ x − y ∣ λ d x , x ∈ R + n + 1 ,

where x = ( x ′ , x n + 1 ) ∈ R + n + 1 , y = ( y ′ , y n + 1 ) ∈ R + n + 1 , 0 < λ < n + 1 , α > − λ 2 n , β > − λ 2 n , α + β ≤ 0 , n + 1 + α + β − λ ≥ 0 .

In this article, we study the following weighted integral system on the half-space R + n + 1 in the special case of α = 0 :

(4) u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 , v ( x ) = ∫ R + n + 1 y n + 1 β g ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 ,

where 0 < λ < n + 1 , β ≤ 0 , β > − 1 p ′ , and n + 1 + β − λ ≥ 0 .

Theorem 1

For n ≥ 1 , let ( u , v ) ∈ C 0 ( R + n + 1 ) × C 0 ( R + n + 1 ) be a pair of positive solutions of (4) and f , g : R + × R + → R + are locally bounded satisfying

  1. f ( s , t ) , g ( s , t ) are strictly increasing in t for fixed s and f ( s , t ) , g ( s , t ) are strictly increasing in s for fixed t ;

  2. there exist p 1 , q 1 ≥ 0 , p 1 + q 1 = 2 ( n + 1 ) + 2 β − λ λ such that f ( s , t ) s − p 1 t − q 1 is non-increasing in t for fixed s and non-increasing in s for fixed t;

  3. there exist p 2 , q 2 ≥ 0 , p 2 + q 2 = 2 ( n + 1 ) + 2 β − λ λ such that g ( s , t ) s − p 2 t − q 2 is non-increasing in t for fixed s and non-increasing in s for fixed t.

Then, one of the following two alternatives holds.

For some x 0 ∈ ∂ R + n + 1 , for any x ∈ ∂ R + n + 1 , u and v must take the following form:

u ( x , 0 ) = c 1 ( d 2 + ∣ x − x 0 ∣ 2 ) − λ 2 , v ( x , 0 ) = c 2 ( d 2 + ∣ x − x 0 ∣ 2 ) − λ 2 .

Furthermore, f ( s , t ) and g ( s , t ) must be the form of

f ( s , t ) = C 1 s p 1 t q 1 and g ( s , t ) = C 2 s p 2 t q 2 ,

where C 1 and C 2 are some positive constants.

The pioneering application of the method of moving spheres can be attributed to Chen and Li [16], Li and Zhu [32], and Padilla [33]. This approach offers a direct method to capture the explicit form of solutions, bypassing the need for prior derivation of radial symmetry when classifying radial solutions. To explore further developments related to the method of moving spheres or moving planes, readers are encouraged to refer to [22,34–47] and the relevant references provided therein. These sources provide valuable insights into the advancements made in this area of research.

Throughout the subsequent analysis, we adopt the symbol C to represent a generic positive constant that could potentially rely on variables such as n , α , β , u , and v . It is important to emphasize that the specific value of this constant might vary from one line of reasoning to another.

2 Proof of Theorem 1

The purpose of this section is to categorize positive solutions for the integral system (4). To simplify the discussion, we introduce some notation. Specifically, for λ > 0 , we denote

B r ( x ) = { y ∈ R n + 1 ∣ ∣ y − x ∣ < r , x ∈ R n + 1 } , B r + ( x ) = { y = ( y 1 , y 2 , … , y n + 1 ) ∈ B r ( x ) ∣ y n + 1 > 0 , x ∈ R n } .

Let z ∈ ∂ R + n + 1 and r > 0 , for all x ∈ R + n + 1 and y ∈ R + n + 1 set

x z , r = r 2 ( x − z ) ∣ x − z ∣ 2 + z and y z , r = r 2 ( y − z ) ∣ y − z ∣ 2 + z ,

then we have

∣ x z , r − y z , r ∣ = r 2 ∣ x − y ∣ ∣ x − z ∣ ∣ y − z ∣ , ∣ y − z ∣ ∣ x − y z , r ∣ = ∣ x − z ∣ ∣ x z , r − y ∣ ,

and

( x z , r ) n = r ∣ x − z ∣ 2 x n .

Define the Kelvin-type transformations

u z , r ( x ) = r ∣ x − z ∣ λ u ( x z , r )

and

v z , r ( x ) = r ∣ x − z ∣ λ u ( x z , r ) .

Lemma 2

Suppose (u, v) is a pair of positive solutions for the integral system (4). For any z ∈ ∂ R + n + 1 and z > 0 , we can establish the following relationship:

(5) u ( x ) − u z , r ( x ) = ∫ B r + ( z ) K ( z , r , y , x ) f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) d y ,

for z ∈ ∂ R + n + 1 . Similarly, we can derive that

(6) v ( x ) − v z , r ( x ) = ∫ B r + ( z ) K ( z , r , y , x ) f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) d y ,

where

K ( z , r , y , x ) = y n + 1 β ∣ x − y ∣ λ − r ∣ x − y ∣ λ y n + 1 β ∣ x − y z , r ∣ λ .

Proof

Without loss of generality, we only prove (5),

(7) u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y = ∫ B r + ( z ) y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y + ∫ R + n + 1 \ B r + ( z ) y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y = ∫ B r + ( z ) y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y + ∫ B r + ( z ) ( y z , r ) n + 1 β f ( u ( y z , r ) , v ( y z , r ) ) ∣ x − y z , r ∣ λ d y z , r = ∫ B r + ( z ) y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y + ∫ B r + ( z ) r ∣ y − z ∣ 2 ( n + 1 ) + 2 β y n + 1 β f ( u ( y z , r ) , v ( y z , r ) ) ∣ x − y z , r ∣ λ d y ,

then we can know that

(8) u z , r ( x ) = r ∣ x − z ∣ λ ∫ B r + ( z ) y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x z , r − y ∣ λ d y + r ∣ x − z ∣ λ ∫ B r + ( z ) r ∣ y − z ∣ 2 ( n + 1 ) + 2 β y n + 1 β f ( u ( y z , r ) , v ( y z , r ) ) ∣ x z , r − y z , r ∣ λ d y = ∫ B r + ( z ) r ∣ y − z ∣ λ y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y + ∫ B r + ( z ) r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ y n + 1 β f ( u ( y z , r ) , v ( y z , r ) ) ∣ x − y ∣ λ d y .

Together with (7) and (8), we have

u ( x ) − u z , r ( x ) = ∫ B r + ( z ) K ( z , r , y , x ) f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) d y .

Similarly, we have

v ( x ) − v z , r ( x ) = ∫ B r + ( z ) K ( z , r , y , x ) g ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ g r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) d y ,

where

□ K ( z , r , y , x ) = y n + 1 β ∣ x − y ∣ λ − r ∣ x − y ∣ λ y n + 1 β ∣ x − y z , r ∣ λ .

Lemma 3

Fix r small for 0 < δ ≪ r , there exists a constant c > 0 , such that

(9) u z , r ( x ) − u ( x ) > 1 , f o r x ∈ B δ + ( z ) ,

and

(10) v z , r ( x ) − v ( x ) > 1 , f o r x ∈ B δ + ( z ) .

Proof

Given that ( u , v ) is a pair of positive solutions for the integral system (4) and f > 0 is strictly increasing on ( 0 , + ∞ ) × ( 0 , + ∞ ) . We can safely assume, without loss of generality, that z = 0 . Under this assumption, it follows that for all ∣ x ∣ ≥ 1 , there exists a positive constant C 1 ¯ > 0 such that the following inequality holds:

u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y ≥ ∫ B 1 2 + ( 0 ) y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y ≥ C ∣ x ∣ λ ∫ B 1 2 + ( 0 ) y n + 1 β f ( u ( y ) , v ( y ) ) d y ≥ C 1 ¯ ∣ x ∣ λ .

Fix r small for 0 < δ ≪ r and x ∈ B δ + ( 0 ) . It can be easily verified that ∣ x 0 , r ∣ takes on large values. As a result, we can deduce that

u 0 , r ( x ) = r ∣ x ∣ λ u ( x 0 , r ) ≥ r ∣ x ∣ λ C 1 ¯ ∣ x 0 , r ∣ λ ≥ C 1 ¯ r λ .

Now, we choose r sufficiently small, such that

u z , r ( x ) − u ( x ) ≥ C 1 ¯ r λ − max x ∈ B δ + ( 0 ) u ( x ) > 1 .

Similar to (9), we can derive that for x ∈ B δ + ( 0 ) , we have

□ v z , r ( x ) − v ( x ) > 1 .

Lemma 4

For z ∈ ∂ R + n + 1 there exists ε 0 ( z ) > 0 . This ensures that for any r ∈ ( 0 , ε 0 ( z ) ] the following condition holds true:

(11) u ( x ) ≤ u z , r ( x ) ∀ x ∈ B r + ( z )

and

(12) v ( x ) ≤ v z , r ( x ) ∀ x ∈ B r + ( z ) .

Proof

For z ∈ ∂ R + n + 1 and r > 0 , define

B u − = { x ∈ B r + ( z ) ∣ u ( x ) > u z , r ( x ) } , B v − = { x ∈ B r + ( z ) ∣ v ( x ) > v z , r ( x ) } .

Now, we will prove that for sufficiently small r , B u − = ∅ and B v − = ∅ . We need to estimate ‖ u ( x ) − u z , r ( x ) ‖ L q ( B u − )  and ‖ v ( x ) − v z , r ( x ) ‖ L q ( B v − ) . At first, estimate ‖ u ( x ) − u z , r ( x ) ‖ L q ( B u − ) .

By the monotonicity of F , we derive that

f r ∣ y − z ∣ − λ u ( y ) , r ∣ y − z ∣ − λ v ( y ) = f r ∣ y − z ∣ − λ u ( y ) , r ∣ y − z ∣ − λ v ( y ) r ∣ y − z ∣ − λ u ( y ) p 1 r ∣ y − z ∣ − λ v ( y ) q 1 r ∣ y − z ∣ − λ u ( y ) p 1 r ∣ y − z ∣ − λ v ( y ) q 1 ≥ r ∣ y − z ∣ ( − λ ) ( p 1 + q 1 ) f ( u ( y ) , v ( y ) ) u p 1 ( y ) v q 1 ( y ) u p 1 ( y ) v q 1 ( y ) = r ∣ y − z ∣ ( − λ ) ( p 1 + q 1 ) f ( u ( y ) , v ( y ) ) .

There are four situations as follows.

(i) If u ( y ) ≥ u z , r ( y ) and v ( y ) ≤ v z , r ( y ) , then by the monotonicity of f and F , we have

(13) f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≥ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v ( y ) u z , r ( y ) u ( y ) .

Since

f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v ( y ) u z , r ( y ) u ( y ) r ∣ y − z ∣ − λ u z , r ( y ) p 1 r ∣ y − z ∣ − λ v ( y ) u z , r ( y ) u ( y ) q 1 ≥ f r ∣ y − z ∣ − λ u ( y ) , r ∣ y − z ∣ − λ v ( y ) r ∣ y − z ∣ − λ u ( y ) p 1 r ∣ y − z ∣ − λ v ( y ) q 1 .

Combining (13), we deduce that

(14) f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≥ f r ∣ y − z ∣ − λ u ( y ) , r ∣ y − z ∣ − λ v ( y ) u z , r ( y ) u ( y ) p 1 + q 1 ≥ r ∣ y − z ∣ ( − λ ) ( p 1 + q 1 ) f ( u ( y ) , v ( y ) ) u z , r ( y ) u ( y ) p 1 + q 1 .

Then, using (14), we have

f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≤ f ( u ( y ) , v ( y ) ) − f ( u ( y ) , v ( y ) ) u z , r ( y ) u ( y ) p 1 + q 1 = f ( u ( y ) , v ( y ) ) u p 1 + q 1 ( y ) ( u p 1 + q 1 ( y ) − u z , r p 1 + q 1 ( y ) ) .

By Lemma 3, we infer that

y ∈ { y ∈ R + n + 1 ∣ u ( y ) ≥ u z , r ( y ) } ∩ { y ∈ R + n + 1 ∣ v ( y ) ≤ v z , r ( y ) } ⊆ B r + ( z ) \ B δ + ( z ) ,

where δ is given in Lemma 3. We observe that f is strictly increasing and u > 0 is continuous. Consequently, there exists a constant C 2 ¯ , which may depend on r and δ satisfying the following condition:

(15) f ( u ( y ) , v ( y ) ) v p 1 + q 1 ( y ) ≤ C 2 ¯ < + ∞ , ∀ y ∈ B r + .

Then, applying the mean value theorem, we deduce from (15) that

f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≤ C 2 ¯ ( u p 1 + q 1 ( y ) − u z , r p 1 + q 1 ( y ) ) ≤ C u p 1 + q 1 − 1 ( y ) ( u ( y ) − u z , r ( y ) ) .

(ii) If u ( y ) ≥ u z , r ( y ) and v ( y ) > v z , r ( y ) , then we have

f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) r ∣ y − z ∣ − λ u z , r ( y ) p 1 r ∣ y − z ∣ − λ v z , r ( y ) q 1 ≥ f r ∣ y − z ∣ − λ u ( y ) , r ∣ y − z ∣ − λ v ( y ) r ∣ y − z ∣ − λ u ( y ) p 1 r ∣ y − z ∣ − λ v ( y ) q 1 ≥ r ∣ y − z ∣ ( − λ ) ( p 1 + q 1 ) f ( u ( y ) , v ( y ) ) .

So we have

f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≤ f ( u ( y ) , v ( y ) ) − f ( u ( y ) , v ( y ) ) u z , r ( y ) u ( y ) p 1 + q 1 v z , r ( y ) v ( y ) p 1 + q 1 ≤ f ( u ( y ) , v ( y ) ) 1 − u z , r ( y ) u ( y ) p 1 + q 1 v z , r ( y ) v ( y ) p 1 + q 1 ≤ f ( u ( y ) , v ( y ) ) 1 − u z , r ( y ) u ( y ) p 1 + q 1 + f ( u ( y ) , v ( y ) ) 1 − v z , r ( y ) v ( y ) p 1 + q 1 = f ( u ( y ) , v ( y ) ) u p 1 + q 1 ( y ) ( u z , r p 1 + q 1 ( y ) − u p 1 + q 1 ( y ) ) + f ( u ( y ) , v ( y ) ) v p 1 + q 1 ( y ) ( v z , r p 1 + q 1 ( y ) − v p 1 + q 1 ( y ) ) ≤ C 3 ¯ ( u p 1 + q 1 ( y ) − u z , r p 1 + q 1 ( y ) ) + C 4 ¯ ( u p 1 + q 1 ( y ) − u z , r p 1 + q 1 ( y ) ) ≤ C [ u p 1 + q 1 − 1 ( y ) ( u ( y ) − u z , r ( y ) ) + v p 1 + q 1 − 1 ( y ) ( v ( y ) − v z , r ( y ) ) ] .

(iii) If u ( y ) < u z , r ( y ) and v ( y ) ≥ v z , r ( y ) , then we can change the roles of u , v in case (i) and obtain

f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≤ C v p 1 + q 1 − 1 ( y ) ( v ( y ) − v z , r ( y ) ) .

(iv) If u ( y ) < u z , r ( y ) and v ( y ) < v z , r ( y ) , then

f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≥ f r ∣ y − z ∣ − λ u ( y ) , r ∣ y − z ∣ − λ v ( y ) ≥ r ∣ y − z ∣ ( − λ ) ( p 1 + q 1 ) ,

we have

f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) ≤ 0 .

Therefore, for every x ∈ B u − , we can infer from cases (i) to (iv) that

u ( x ) − u z , r ( x ) = ∫ B r + ( z ) K ( z , r , y , x ) f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) d y ≤ C ∫ B u − K ( z , r , y , x ) u p 1 + q 1 − 1 ( y ) ( u ( y ) − u z , r ( y ) ) d y + ∫ B v − K ( z , r , y , x ) v p 1 + q 1 − 1 ( y ) ( v ( y ) − v z , r ( y ) ) d y ≤ C ∫ B u − y n + 1 β ∣ x − y ∣ λ u p 1 + q 1 − 1 ( y ) ( u ( y ) − u z , r ( y ) ) d y + ∫ B v − y n + 1 β ∣ x − y ∣ λ v p 1 + q 1 − 1 ( y ) ( v ( y ) − v z , r ( y ) ) d y .

Similarly, we have

v ( x ) − v z , r ( x ) = ∫ B r + ( z ) K ( z , r , y , x ) f ( u ( y ) , v ( y ) ) − r ∣ y − z ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y − z ∣ − λ u z , r ( y ) , r ∣ y − z ∣ − λ v z , r ( y ) d y ≤ C ∫ B u − y n + 1 β ∣ x − y ∣ λ u p 2 + q 2 − 1 ( y ) ( u ( y ) − u z , r ( y ) ) d y + ∫ B v − y n + 1 β ∣ x − y ∣ λ v p 2 + q 2 − 1 ( y ) ( v ( y ) − v z , r ( y ) ) d y .

By the HLS inequality and the Hölder inequality, we have

‖ u − u z , r ‖ L p 1 + q 1 + 1 ( B u − ) ≤ C ‖ u ‖ L 2 ( n + 1 ) λ ( B u − ) p 1 + q 1 − 1 ‖ u − u z , r ‖ L p 1 + q 1 + 1 ( B u − ) + ‖ v ‖ L 2 ( n + 1 ) λ ( B v − ) p 2 + q 2 − 1 ‖ v − v z , r ‖ L p 2 + q 2 + 1 ( B v − )

and

‖ v ( x ) − v z , r ( x ) ‖ L p 2 + q 2 ( B v − ) ≤ C ‖ u ‖ L 2 ( n + 1 ) λ ( B u − ) p 1 + q 1 − 1 ‖ u − u z , r ‖ L p 1 + q 1 + 1 ( B u − ) + ‖ v ‖ L 2 ( n + 1 ) λ ( B v − ) p 2 + q 2 − 1 ‖ v − v z , r ‖ L p 2 + q 2 + 1 ( B v − ) .

Since 2 ( n + 1 ) λ > 1 and ( u , v ) ∈ C 0 ( R + n + 1 ) × C 0 ( R + n + 1 ) , there exists ε 0 ( z ) such that for 0 < r < ε 0 ( z ) ,

C ‖ u ( x ) ‖ L 2 ( n + 1 ) λ ( B u − ) p 1 + q 1 − 1 ≤ 1 4 , C ‖ v ( x ) ‖ L 2 ( n + 1 ) λ ( B v − ) p 2 + q 2 − 1 ≤ 1 4 ,

we deduce that

‖ u ( x ) − u z , r ( x ) ‖ L p 1 + q 1 + 1 ( B u − ) = ‖ v ( x ) − v z , r ( x ) ‖ L p 2 + q 2 + 1 ( B v − ) = 0 ,

i.e., B u − and B v − have measure 0 and hence empty.

This completes the proof of Lemma 4.□

Continuing the moving sphere procedure, we iterate as long as conditions (11) and (12) hold. For each fixed z ∈ ∂ R + n + 1 , define

(16) r ¯ ( z ) = sup { r > 0 ∣ u ≤ u z , r in B μ + ( z ) , v ≤ v z , r in B μ + ( z ) , ∀ 0 < μ ≤ r } .

By Lemma 4, r ¯ ( z ) is well defined and 0 < r ¯ ( z ) ≤ + ∞ for any z ∈ ∂ R + n + 1 .

To establish our proof, we require the following lemma, which plays a crucial role.

Lemma 5

If r ¯ ( z ) < ∞ for some z ¯ ∈ ∂ R + n + 1 , then

(17) u ( x ) = u z ¯ , r ¯ ( x ) , f o r x ∈ B r ¯ ( z ¯ ) + ( z ¯ ) ,

and

(18) v ( x ) = v z ¯ , r ¯ ( x ) , f o r x ∈ B r ¯ ( z ¯ ) + ( z ¯ ) .

Proof

Without loss of generality, assuming z ¯ = 0 , for convenience in notation, we set r ¯ = r ¯ ( 0 ) . Consider the opposite scenario where u 0 , r ¯ − u ≥ 0 in B r ¯ + ( 0 ) and v 0 , r ¯ − v ≥ 0 in B r ¯ + ( 0 ) but at least one of them is not identically zero. Without loss of generality, let us assume that v 0 , r ¯ − v ≥ 0 but v 0 , r ¯ − v is not identically zero in B r ¯ + ( 0 ) . We will now derive a contradiction based on the definition of z ¯ .

We first show that

(19) u 0 , r ¯ ( x ) − u ( x ) > 0 , ∀ x ∈ B r ¯ + ( 0 ) ,

and

(20) v 0 , r ¯ ( x ) − v ( x ) > 0 , ∀ x ∈ B r ¯ + ( 0 ) .

Indeed, if there exists a point y 0 ∈ B r ¯ + ( 0 ) such that v 0 , r ¯ ( y 0 ) − v ( y 0 ) > 0 , we can infer from the continuity of v that there exist small positive numbers η and c 0 such that

(21) B η + ( 0 ) ⊂ B r ¯ + ( 0 ) and v 0 , r ¯ ( y ) − v ( y ) ≥ c 0 > 0 , ∀ y ∈ B η + ( 0 ) .

For any y ∈ B η + ( y 0 ) , using (16) and (20) and the strict monotonicity of f , we can deduce that

f r ∣ y ∣ − λ u 0 , r ¯ ( y ) , r ∣ y ∣ − λ v 0 , r ¯ ( y ) > f r ∣ y ∣ − λ u 0 , r ¯ ( y ) , r ∣ y ∣ − λ v ( y )

(22) ≥ f r ∣ y ∣ − λ u ( y ) , r ∣ y ∣ − λ v ( y ) ≥ r ∣ y ∣ ( − λ ) ( p 1 + q 1 ) f ( u ( y ) , v ( y ) ) u p 1 ( y ) v q 1 ( y ) u p 1 ( y ) v q 1 ( y ) = r ∣ y ∣ ( − λ ) ( p 1 + q 1 ) f ( u ( y ) , v ( y ) ) .

For any x ∈ B r ¯ + ( 0 ) , since K ( 0 , r ¯ , y , x ) > 0 , one can derive from (5) and (21) that

(23) u 0 , r ¯ ( x ) − u ( x ) = ∫ B r + ( 0 ) K ( 0 , r ¯ , y , x ) r ∣ y ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y ∣ − λ u 0 , r ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) − f ( u ( y ) , v ( y ) ) d y ≥ ∫ B η + ( y 0 ) K ( 0 , r ¯ , y , x ) r ∣ y ∣ 2 ( n + 1 ) + 2 β − λ f r ∣ y ∣ − λ u 0 , r ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) − f ( u ( y ) , v ( y ) ) d y > 0 .

Therefore, we obtain equation (19). Equation (6) and the positivity of K ( 0 , r ¯ , y , x ) lead to the immediate derivation of equation (20). Moreover, equations (22) and (23) imply that there exists a 0 < ι < r ¯ small enough, and for any x ∈ B ι + ( 0 ) , we have

(24) u 0 , r ¯ ( x ) − u ( x ) ≥ c 1 > 0 ,

which implies that u 0 , r ¯ ( x ) − u ( x ) has a strictly positive lower bound in a small neighborhood of 0. Select δ 1 ∈ ( 0 , min { δ 2 , r ¯ 2 } ) based on the value of δ provided by Lemma 3. By (19) and (24), we define

(25) m 1 = inf x ∈ B r ¯ − δ 1 + ( 0 ) \ { 0 } ¯ ( u 0 , r ¯ − u ) ( x ) > 0 .

Then, by the continuity of u 0 , r ¯ − u with respect to r , we can find a positive constant δ 2 satisfying 0 < δ 2 < δ 1 0 6 . This ensures that for every r ∈ [ r ¯ , r ¯ + δ 2 ) , we have

(26) u 0 , r ¯ ( x ) − u ( x ) ≥ m 1 2 > 0 , ∀ x ∈ B r ¯ − δ 1 + ( 0 ) ¯ \ { 0 } .

Then, our goal is to demonstrate the existence of a positive constant m 2 such that

(27) v 0 , r ¯ ( x ) − v ( x ) ≥ m 2 > 0 , ∀ x ∈ B r ¯ − δ 1 + ( 0 ) ¯ \ { 0 } .

By (6), for any x ∈ B r ¯ − δ 1 + ( 0 ) ¯ \ { 0 } , one can deduce that

(28) v 0 , r ( x ) − v ( x ) = ∫ B r + ( 0 ) K ( 0 , r , y , x ) r ∣ y ∣ 2 ( n + 1 ) + 2 β − λ g r ∣ y ∣ α − n u 0 , r ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) − g ( u ( y ) , v ( y ) ) d y = ∫ B r ¯ − δ 1 + ( 0 ) K ( 0 , r , y , x ) r ∣ y ∣ 2 ( n + 1 ) + 2 β − λ g r ∣ y ∣ − λ u 0 , r ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) − g ( u ( y ) , v ( y ) ) d y + ∫ B r + ( 0 ) \ B r ¯ − δ 1 + ( 0 ) K ( 0 , r , y , x ) r ∣ y ∣ 2 ( n + 1 ) + 2 β − λ g r ∣ y ∣ − λ u 0 , r ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) − g ( u ( y ) , v ( y ) ) d y = I 1 ( x ) + I 2 ( x ) .

We first take care of I 1 ( x ) . For any y ∈ B r ¯ − δ 1 + ( 0 ) ¯ \ { 0 } , it follows from (26) and the monotonicity of g and G that

g r ∣ y ∣ − λ u 0 , r ( y ) , r ∣ y − z ∣ − λ v 0 , r ( y )

(29) > g r ∣ y ∣ − λ u ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) ≥ g r ∣ y ∣ − λ u ( y ) , r ∣ y ∣ − λ v ( y ) ≥ r ∣ y ∣ ( − λ ) ( p 1 + q 1 ) g ( u ( y ) , v ( y ) ) u p 1 ( y ) v q 1 ( y ) u p 1 ( y ) v q 1 ( y ) = r ∣ y ∣ ( − λ ) ( p 1 + q 1 ) g ( u ( y ) , v ( y ) ) .

By (26) and (29), along with the strict monotonicity of g , we can conclude that there exists a positive constant m 2 depending on n , α , r , and r ¯ such that

I 1 ( x ) ≥ 2 m 2 > 0 .

To estimate I 2 ( x ) , we consider y ∈ B r + ( 0 ) \ B r ¯ − δ 1 + ( 0 ) ¯ . We can use the fact that g is strictly increasing and u is continuous to establish the existence of a uniform constant C on compact set B r + ( 0 ) \ B r ¯ − δ 1 + ( 0 ) ¯ . This constant depends n , β , r , and r ¯ such that

K ( 0 , r ¯ , y , x ) r ∣ y ∣ 2 ( n + 1 ) + 2 β − λ g r ∣ y ∣ − λ u 0 , r ( y ) , r ∣ y ∣ − λ v 0 , r ( y ) − g ( u ( y ) , v ( y ) ) ≤ C .

By combining the previous estimate with equation (28), we can show that for sufficiently small δ 1 > 0 ,

v 0 , r ¯ ( x ) − v ( x ) = I 1 ( x ) + I 2 ( x ) ≥ 2 m 2 − C 5 ¯ [ r n − 1 − ( r ¯ − δ 1 ) n − 1 ] = 2 m 2 − C 6 ¯ [ r − ( r ¯ − δ 1 ) ] ≥ 2 m 2 − C 6 ¯ [ δ 1 + δ 2 ] ≥ m 2 > 0 .

Therefore, (25) is established. Thus, we can use the narrow region B r + ( 0 ) \ B r ¯ − δ 1 + ( 0 ) instead of B r + ( 0 ) \ B r − δ 1 + ( 0 ) in Lemma 4, following a similar proof process as in Lemma 4. Then, we conclude that B u − = B v − = ∅ for all r ∈ [ r ¯ , r ¯ + δ 2 ] , i.e.,

u 0 , r ( x ) − u ( x ) ≥ 0 , ∀ x ∈ B r + ( 0 ) ,

and

v 0 , r ( x ) − v ( x ) ≥ 0 , ∀ x ∈ B r + ( 0 ) ,

which contradicts the definition of r ¯ . As a consequence, in the case 0 < r ¯ < + ∞ , we must have

u 0 , r ¯ ( x ) = u ( x ) , ∀ x ∈ B r ¯ + ( 0 ) ,

and

v 0 , r ¯ ( x ) = v ( x ) , ∀ x ∈ B r ¯ + ( 0 ) .

This concludes the proof of Lemma 5.□

Additionally, we must consider about the limiting radius r ¯ ( z ) .

Lemma 6

If r ¯ ( z ¯ ) = + ∞ for some z ¯ ∈ ∂ R + n + 1 , then r ¯ ( z ) = + ∞ for all z ∈ ∂ R + n + 1 .

Proof

Since r ¯ ( z ¯ ) = + ∞ , recalling the definition of r ¯ , we derive

u z ¯ , r ( x ) ≥ u ( x ) , ∀ x ∈ B r + ( z ¯ ) , ∀ 0 < r < + ∞ ,

i.e.,

u ( x ) ≥ u z ¯ , r ( x ) , ∀ ∣ x − ( z ¯ ) ∣ ≥ r , ∀ 0 < r < + ∞ .

It follows immediately that

(30) lim ∣ x ∣ → ∞ ∣ x ∣ λ u ( x ) = + ∞ .

On the other hand, if we assume r ¯ ( z ) < + ∞ for some z ∈ ∂ R + n + 1 , then by Lemma 5, we arrive at

lim ∣ x ∣ → ∞ ∣ x ∣ λ u ( x ) = lim ∣ x ∣ → ∞ ∣ x ∣ λ u z ¯ , r ( z ) ( x ) = ( r ( z ) ) λ u ( z ) < + ∞ ,

which contradicts (30).

Thus, we have completed the proof of Lemma 6.□

Now, we are prepared to present a proof of Theorem 1.

Proof

We will proceed with the proof by considering two different possible cases.

Case (i). r ¯ ( z ) < ∞ , for all z ∈ ∂ R + n .

From Lemma 5, we infer that

(31) u ( x ) = u z ¯ , r ¯ ( z ¯ ) ( x ) , ∀ x ∈ B r ¯ + ( z ¯ ) .

By exploiting the conformal invariance property of (4) and applying a calculus lemma (Lemma 5.8 in Li [20]) along with (31), we can infer that for any x ∈ ∂ R + n + 1 ,

u ( x , 0 ) = c 1 ( d 2 + ∣ x − x 0 ∣ 2 ) − λ 2 ,

for c 1 > 0 , d > 0 , x 0 ∈ ∂ R + n + 1 . Similarly, for any x ∈ ∂ R + n + 1 , we have

v ( x , 0 ) = c 2 ( d 2 + ∣ x − x 0 ∣ 2 ) − λ 2 ,

for some c 2 > 0 , d > 0 and x 0 ∈ ∂ R + n + 1 . Now, we show that F and G are constants. It follows from (5), (17), and (18) that

u z , r ¯ ( x ) − u ( x ) = ∫ B r ¯ + ( z ) K ( 0 , r ¯ , y , x ) r ¯ ∣ y ∣ 2 ( n + 1 ) + 2 β − λ f r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) , r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) − f ( u ( y ) , v ( y ) ) d y = ∫ B r ¯ + ( z ) K ( 0 , r ¯ , y , x ) f r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) , r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) p 1 r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) q 1 u 0 , r ¯ p 1 ( y ) v 0 , r ¯ q 1 ( y ) d y − ∫ B r ¯ + ( z ) K ( 0 , r ¯ , y , x ) f ( u ( y ) , v ( y ) ) u p 1 ( y ) v q 1 ( y ) u p 1 ( y ) v q 1 ( y ) d y = ∫ B r ¯ + ( z ) K ( 0 , r ¯ , y , x ) f r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) , r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) p 1 r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) q 1 − f ( u ( y ) , v ( y ) ) u p 1 ( y ) v q 1 ( y ) ⋅ u p 1 ( y ) v q 1 ( y ) d y ≥ 0 .

In the aforementioned derivation, we have made use of the positivity of K ( 0 , r ¯ , y , x ) and the monotonicity of F . Hence, for any x ∈ B r ¯ + ( z ) , we obtain

f r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) , r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) r ¯ ∣ y ∣ − λ u 0 , r ¯ ( y ) p 1 r ¯ ∣ y ∣ − λ v 0 , r ¯ ( y ) q 1 = f ( u ( y ) , v ( y ) ) u p 1 ( y ) v q 1 ( y ) .

Therefore, for some positive constant C 1 , we have

F ( t ) = f ( s , t ) s p 1 t q 1 = C 1 , t ∈ ( 0 , max x ∈ R + n + 1 v ( x ) ] .

In a similar manner, we can deduce that for some positive constant C 2 ,

G ( t ) = f ( t ) s p 2 t q 2 = C 2 , t ∈ ( 0 , max x ∈ R + n + 1 u ( x ) ] .

This completes our proof of Theorem 1.

Case (ii). By C a s e ( i ) , we know that r ( x ) = ∞ for all x ∈ R n . Then, u ≡ C 1 ˜ , v ≡ C 2 ˜ ( C 1 ˜ a n d C 2 ˜ are some constants) and f ( C 1 ˜ , C 2 ˜ ) = 0 , g ( C 1 ˜ , C 2 ˜ ) = 0 . This contradicts f , g : R + × R + → R + ; thus, this case is impossible.□

3 Conclusion

In this article, we study a weighted integral system and directly obtain the classification of positive solutions in the upper half-space using the method of moving spheres.

Acknowledgements

The authors would like to express their sincere thanks to Professor Zhao Liu for his valuable suggestions and useful comments, from which the authors have benefited on this manuscript, and the authors express their gratitude to the editors and reviewers for their serious and responsible comments and pertinent suggestions.

  1. Funding information: The authors have been supported by the Science and Technology Project of Education Department of Jiangxi Province (GJJ2201354, GJJ218419), the Teaching Reform Project of Jiangxi Science and Technology Normal University (XJJG-2022-50-07), the Doctoral Science Foundation, Jiangxi Science and Technology Normal University (2022BSQD13), and Basic Education Research Project in Jiangxi Province (SZUJKSX2024-1022).

  2. Author contributions: All authors conceived the study, drafted the manuscript, and approved the final manuscript.

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

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Received: 2023-11-13
Revised: 2024-08-03
Accepted: 2024-08-07
Published Online: 2024-09-16

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

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  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  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”
https://doi.org/10.1515/math-2024-0058
Keywords for this article
method of moving spheres; weighted integral system
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Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  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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