Home Mathematics Bubbles clustered inside for almost-critical problems
Article Open Access

Bubbles clustered inside for almost-critical problems

  • Mohamed Ben Ayed and Khalil El Mehdi EMAIL logo
Published/Copyright: June 5, 2025
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

We investigate the existence of blowing-up solutions of the following almost-critical problem:

− Δ u + V ( x ) u = u p − ε , u > 0 in Ω , u = 0 on ∂ Ω ,

where Ω is a bounded regular domain in R n , n ≥ 4 , ε is a small positive parameter, p + 1 = ( 2 n ) ∕ ( n − 2 ) is the critical Soblolev exponent, and the potential V is a smooth positive function. We find solutions that exhibit bubbles clustered inside as ε goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional.

Keywords: partial differential equations; variational analysis; nonlinear analysis; critical Sobolev exponent
MSC 2010: 35A15; 35J20; 35J25

1 Introduction and main results

In this article, we study the following almost-critical problem:

(1) ( P V , ε ) : − Δ u + V u = u p − ε , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω ,

where Ω is a bounded regular domain in R n , n ≥ 4 , p + 1 = ( 2 n ) ∕ ( n − 2 ) is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L q ( Ω ) , the potential V is a C 3 positive function on Ω ¯ , and ε is a small positive parameter.

The problem in the form of ( P V , ε ) arises in various physical models, such as quantum transport and non-relativistic Newtonian gravity, as detailed in [1–3] and their associated references. It is also connected to the Yamabe problem in differential geometry, as discussed, for example, in [4] and references therein.

In the critical case, where ε = 0 , it is well established that the existence of solutions to problem ( P V , ε ) is influenced by the geometry of the domain, the characteristics of the potential V , and the dimension n . For instance, when V is constant and the domain is star-shaped, there are no solutions to the given problem. Due to the vast amount of research on this subject, we will only mention the seminal works by Brezis and Nirenberg [5] and Bahri and Coron [6].

In the subcritical case where ε > 0 , proving the existence of a solution to the problem ( P V , ε ) is relatively simple. This can be shown by observing that the infimum

inf ∫ Ω ( ∣ ∇ u ∣ 2 + V u 2 ) : u ∈ H 0 1 ( Ω ) and ∫ Ω ∣ u ∣ p + 1 − ε = 1

is achieved, thanks to the compactness of the embedding H 0 1 ( Ω ) ↪ L p + 1 − ε ( Ω ) .

In [7], solutions of ( P V , ε ) were constructed that concentrate, as ε → 0 , at interior blow-up points, forming isolated bubbles. By bubble here, we refer to the functions defined by

δ a , λ ( x ) ≔ c 0 λ ( n − 2 ) ∕ 2 ( 1 + λ 2 ∣ x − a ∣ 2 ) ( n − 2 ) ∕ 2 , with c 0 ≔ [ n ( n − 2 ) ] n − 2 4 , a ∈ R n , λ > 0 ,

which are the only solutions to equation [8]

− Δ u = u p , u > 0 in R n .

In this article, we focus on the construction of interior bubbling solutions of ( P V , ε ) with clustered bubbles at critical points of V . These solutions reveal a new phenomenon for positive solutions, namely, the existence of non-simple blow-up points in the interior for the subcritical problem with Dirichlet boundary conditions in general domains. This phenomenon has been observed for sign-changing solutions on certain symmetric domains [9], as well as for positive solutions on the ball [10]. It has also been noted in the setting of non-locally conformally compact Riemannian manifolds [11,12]. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains.

In order to formulate our results, we need to introduce some notation. Let b be a non-degenerate critical point of V and N ∈ N , we define the following function:

(2) ℱ b , N ( z 1 , … , z N ) ≔ ∑ i = 1 N D 2 V ( b ) ( z i , z i ) − ∑ 1 ≤ j ≠ i ≤ N 1 ∣ z j − z i ∣ n − 2 .

The aim of our first result is to construct interior bubbling solutions with clustered bubbles at a critical point of V , Namely, we have

Theorem 1.1

Let n ≥ 4 , N ≥ 2 , and b ∈ Ω be a non-degenerate critical point of V. Assume that the function ℱ b , N has a non-degenerate critical point ( z ¯ 1 , … , z ¯ N ) . Then, there exists a small positive real ε 0 such that for any ε ∈ ( 0 , ε 0 ] , problem ( P V , ε ) has a solution u ε , b with the following property:

u ε , b = ∑ i = 1 N α i , ε δ a i , ε , λ i , ε + v ε ,

with

(3) ‖ v ε ‖ < c ε ,

(4) ∣ α i , ε − 1 ∣ ≤ ε ln 2 ε ∀ 1 ≤ i ≤ N ,

(5) 1 c ≤ ln σ n λ i , ε ε λ i , ε 2 ≤ c ∀ 1 ≤ i ≤ N ,

(6) ∣ a i , ε − b + η ( ε ) σ z ¯ i ∣ ≤ η ( ε ) η 0 ∀ 1 ≤ i ≤ N ,

where η 0 is a small positive constant, σ 4 = 1 , σ n = 0 for n ≥ 5 , c is a positive constant, η ( ε ) and σ are defined in (54).

In addition, if for each N, ℱ b , N has a non-degenerate critical point, then, problem ( P V , ε ) has an arbitrary number of non-constant distinct solutions, provided that ε is small.

Remark 1.2

  1. Note that for N = 1 , the existence of solution blowing up at a critical point of V with only one bubble has already been proved in [7].

  2. One can readily construct examples of potentials V for which the function ℱ b , N , defined in (2), possesses a non-degenerate critical point. For instance, let us fix a point b ∈ Ω , denote R = diam ( Ω ) , and choose a constant γ > 0 such that 2 n − 1 γ n = n − 2 . Define the points z ¯ 1 = ( γ , 0 ) ∈ R × R n − 1 and z ¯ 2 = − z ¯ 1 , and consider the potential function

    V ( x ) = R 2 − 1 2 ( x 1 − b 1 ) 2 + 1 2 ∑ i = 2 n ( x i − b i ) 2 .

    Then, the associated functional ℱ b , 2 , defined for z 1 = ( z 11 , z 12 ) , z 2 = ( z 21 , z 22 ) ∈ R × R n − 1 , takes the form

    ℱ b , 2 ( z 1 , z 2 ) = − z 11 2 − z 21 2 + ∣ z 12 ∣ 2 + ∣ z 22 ∣ 2 − 2 ( ( z 11 − z 21 ) 2 + ∣ z 12 − z 22 ∣ 2 ) n − 2 2 .

    A straightforward computation shows that the point z ¯ = ( z ¯ 1 , z ¯ 2 ) is a non-degenerate critical point of ℱ b , 2 .

Our second result addresses the case of multiple interior blow-up points with clustered bubbles. More precisely, we prove:

Theorem 1.3

Let n ≥ 4 , b 1 , … , b k ∈ Ω be non-degenerate critical points of V and N 1 , … , N k be non-zero natural integers. For each i ∈ { 1 , … , k } such that N i ≥ 2 , assume that the function ℱ b i , N i has a non-degenerate critical point ( z ¯ 1 i , … , z ¯ N i i ) . Then, there exists a small positive real ε 0 such that for any ε ∈ ( 0 , ε 0 ] , problem ( P V , ε ) has a solution u ε , b 1 , … , b k satisfying

u ε , b 1 , … , b k = ∑ i = 1 N 1 α 1 , i , ε δ a 1 , i , ε , λ 1 , i , ε + … + ∑ i = 1 N k α k , i , ε δ a k , i , ε , λ k , i , ε + v ε ,

with ‖ v ε ‖ ≤ c ε and for each l ≤ k the coefficients α l , i , ε , the speeds λ l , i , ε , and the points a l , i , ε satisfy the properties (4), (5), and (6), respectively.

The proof of our results relies on refined asymptotic estimates of the gradient of the associated Euler-Lagrange functional in the neighborhood of bubbles. The goal is to determine the equilibrium conditions satisfied by the concentration parameters. These balancing conditions are derived by testing the equation with vector fields that represent the dominant terms of the gradient relative to the concentration parameters. Analyzing these conditions provides all the necessary information to establish our results.

The rest of this article is organized as follows: in Section 2, we introduce the parameterization of the neighborhood of bubbles, provide a precise estimate of the infinite-dimensional part, and carry out a delicate asymptotic expansion of the gradient of the associated Euler-Lagrange functional. Section 3 is dedicated to proving our results. Finally, in Section 4, we present several estimates that are referenced throughout this article.

2 Analytical framework

Problem ( P V , ε ) has a variational structure, and its Euler-Lagrange functional is defined on H 0 1 ( Ω ) by

(7) I V , ε ( u ) ≔ 1 2 ∫ Ω ∣ ∇ u ∣ 2 + 1 2 ∫ Ω V u 2 − 1 p + 1 − ε ∫ Ω ∣ u ∣ p + 1 − ε with p ≔ n + 2 n − 2 .

In the sequel, we will use the following scalar product and its corresponding norm defined by

(8) ⟨ v , w ⟩ ≔ ∫ Ω ∇ v ⋅ ∇ w + ∫ Ω V v w ; ‖ w ‖ 2 ≔ ∫ Ω ∣ ∇ w ∣ 2 + ∫ Ω V w 2 .

Note that, since V is a C 0 -positive function on Ω ¯ , we deduce that this norm is equivalent to the two norms ∣ ∣ ⋅ ∣ ∣ 0 and ∣ ∣ ⋅ ∣ ∣ 1 of H 0 1 ( Ω ) and H 1 ( Ω ) , respectively.

We start by giving some estimates concerning the approximate solutions for ( P V , ε ) . For a ∈ Ω and λ > 0 , let π δ a , λ the projection defined by

− Δ π δ a , λ + V π δ a , λ = δ a , λ n + 2 n − 2 , in Ω , π δ a , λ = 0 , on ∂ Ω ,

and

θ a , λ ≔ δ a , λ − π δ a , λ .

These functions are introduced in [7], and they satisfy, for n ≥ 4 ,

(9) 0 ≤ π δ a , λ ≤ δ a , λ ; λ ∂ π δ a , λ ∂ λ ≤ n + 2 2 π δ a , λ ; 1 λ ∂ π δ a , λ ∂ a j ≤ n + 2 2 π δ a , λ , in Ω ,

(10) 0 ≤ θ a , λ ≤ δ a , λ ; λ ∂ θ a , λ ∂ λ ≤ n − 2 2 θ a , λ ; 1 λ ∂ θ a , λ ∂ a j ≤ n − 2 2 θ a , λ , in Ω ,

(11) θ a , λ ≤ c R a , λ 1 δ a , λ ; 1 λ ∂ θ a , λ ∂ a j ≤ c R a , λ 2 δ a , λ , in Ω 0 ≔ { x ∈ Ω : d ( x , ∂ Ω ) ≥ d 0 } ,

(12) with R a , λ 1 ( x ) ≔ ln σ n ( λ ) λ − 2 + ∣ x − a ∣ 2 ∣ ln ∣ x − a ∣ ∣ σ n ; R a , λ 2 ( x ) ≔ λ − 2 + λ − 1 ∣ x − a ∣ ,

where d 0 is any fixed small positive constant, σ 4 ≔ 1 , σ n ≔ 0 for n ≥ 5 , and a j is the j th-component of a .

For the proof of these facts, the interested reader is referred to [7].

Now, we introduce the parameterization of the neighborhood of bubbles. Let N ∈ N and μ > 0 be a small real. We define the following set:

(13) O ( N , μ ) ≔ { ( α , a , λ ) ∈ ( 0 , ∞ ) N × Ω N × ( μ − 1 , ∞ ) N : ∣ α i − 1 ∣ < μ ; d ( a i , ∂ Ω ) > 2 d 0 ; ε ln λ i < μ ; ∀ i and ε i j < μ ∀ i ≠ j } ,

where ε i j is defined by

(14) ε i j ≔ λ i λ j + λ j λ i + λ i λ j ∣ a i − a j ∣ 2 ( 2 − n ) ∕ 2 .

Observe that, since Ω is bounded, for each b ∈ Ω and λ > 0 satisfying ε ln λ is small, it follows that

(15) δ b , λ − ε = c 0 − ε λ − ε ( n − 2 ) ∕ 2 1 + n − 2 2 ε ln ( 1 + λ 2 ∣ x − b ∣ 2 ) + O ( ε 2 ln 2 ( 1 + λ 2 ∣ x − b ∣ 2 ) ) = 1 + o ( 1 ) .

For each ( α , a , λ ) ∈ O ( N , μ ) and v ∈ E a , λ ⊥ , we associate a function

(16) u ≔ ∑ i = 1 N α i ( π δ a i , λ i ) + v ≔ u ̲ + v , where

(17) E a , λ ≔ span π δ a i , λ i , ∂ ( π δ a i , λ i ) ∂ λ i , ∂ ( π δ a i , λ i ) ∂ a i , j , i ∈ { 1 , … , N } , j ∈ { 1 , … , n } .

We note that the orthogonality is taken with respect to the scalar product defined in (8).

In the sequel, we need to study the functional I V , ε and to find some positive critical points u having form (16). We start by studying the v -part of u .

2.1 Estimate of the infinite-dimensional part

Let ( α , a , λ ) ∈ O ( N , μ ) , v ∈ E a , λ ⊥ and u be defined in (16). In this section, we are going to study the v -part of u . To this aim, we need to expand I V , ε with respect to v .

Observe that, for b 1 , b 2 ∈ R , and γ > 2 , it holds

(18) ∣ b 1 + b 2 ∣ γ − ∣ b 1 ∣ γ − γ ∣ b 1 ∣ γ − 2 b 1 b 2 − 1 2 γ ( γ − 1 ) ∣ b 1 ∣ γ − 2 b 2 2 ≤ c ∣ b 1 ∣ γ − 3 ∣ b 2 ∣ 3 + c ∣ b 2 ∣ γ , if γ > 3 , c ∣ b 2 ∣ γ , if γ ≤ 3 .

Thus, using the fact that ⟨ v , u ̲ ⟩ = 0 , we obtain

(19) I V , ε ( u ) = I V , ε ( u ̲ ) − f ε ( v ) + 1 2 Q ε ( v ) + o ( ∣ ∣ v ∣ ∣ 2 ) , where

(20) f ε ( v ) ≔ ∫ Ω ( u ̲ ) p − ε v and Q ε ( v ) ≔ ∣ ∣ v ∣ ∣ 2 − ( p − ε ) ∫ Ω ( u ̲ ) p − ε − 1 v 2 .

Note that Q ε is a positive definite quadratic form (see Proposition 3.1 of [7]) and the linear form f ε satisfies:

Lemma 2.1

Let ( α , a , λ ) ∈ O ( N , μ ) . Then, for each v ∈ E a , λ ⊥ , we have

∣ f ε ( v ) ∣ ≤ c ∥ v ∥ ε + ∑ 1 ≤ i ≤ N T 2 ( λ i ) + ∑ i ≠ j T 3 ( ε i j ) , w h e r e T 2 ( λ ) ≔ ( λ − 2 i f n ≥ 5 ; λ − 2 ln λ i f n = 4 ; λ − 1 i f n = 3 ) , T 3 ( t ) ≔ ( t ( n + 2 ) ∕ ( 2 ( n − 2 ) ) ( ln t − 1 ) ( n + 2 ) ∕ 2 n if n ≥ 6 , t if n ≤ 5 ) .

Proof

First, observe that, for β i > 0 and γ > 1 , it holds

(21) ∑ i = 1 N β i γ − ∑ i = 1 N β i γ ≤ c ∑ i ≠ j ( β i β j ) γ ∕ 2 , if γ ≤ 2 , c ∑ i ≠ j β i γ − 1 β j , if γ > 2 .

Thus, we obtain

(22) f ε ( v ) = ∑ i = 1 N α i p − ε ∫ Ω ( π δ a i , λ i ) p − ε v + ∑ i ≠ j O ∫ Ω [ ( π δ a i , λ i ) ( π δ a j , λ j ) ] p − ε 2 ∣ v ∣ , if n ≥ 6 , O ∫ Ω ( π δ a i , λ i ) p − ε − 1 ( π δ a j , λ j ) ∣ v ∣ , if n ≤ 5 .

Observe that, using (9) and (15), we obtain

(23) ∫ Ω [ ( π δ a i , λ i ) ( π δ a j , λ j ) ] p − ε 2 ∣ v ∣ ≤ ∫ Ω ( δ a i , λ i δ a j , λ j ) p 2 ∣ v ∣ ≤ ∫ Ω ( δ a i , λ i δ a j , λ j ) n n − 2 ( n + 2 ) ∕ 2 n ∫ Ω ∣ v ∣ 2 n n − 2 ( n − 2 ) ∕ ( 2 n ) .

Note that estimate (E2) of [13] gives us

(24) ∫ R n ( δ a i , λ i δ a j , λ j ) n ∕ ( n − 2 ) ≤ c ε i j n n − 2 ln ( ε i j − 1 ) ,

and using the embedding theorem of H 0 1 ( Ω ) ↪ L 2 n n − 2 ( Ω ) , we derive that

(25) ∫ Ω [ ( π δ a i , λ i ) ( π δ a j , λ j ) ] p − ε 2 ∣ v ∣ ≤ c ε i j n + 2 2 ( n − 2 ) ( ln ε i j − 1 ) n + 2 2 n ‖ v ‖ .

In the same way, for n ≤ 5 (which implies p − 1 ≔ 4 n − 2 > 1 ), using Lemma 2.2 of [14], we have

(26) ∫ Ω ( π δ a i , λ i ) p − ε − 1 ( π δ a j , λ j ) ∣ v ∣ ≤ ∫ Ω δ a i , λ i p − 1 δ a j , λ j ∣ v ∣ ≤ c ‖ v ‖ ∫ Ω δ a i , λ i ( p − 1 ) 2 n n + 2 δ a j , λ j 2 n n + 2 n + 2 2 n ≤ c ‖ v ‖ ε i j .

It remains to estimate the first integral in (22). Observe that, for 0 < b 2 < b 1 and γ > 1 , it holds

(27) ( b 1 − b 2 ) γ = b 1 γ + O ( b 1 γ − 1 b 2 ) .

Thus, we obtain

∫ Ω ( π δ a i , λ i ) p − ε v = ∫ Ω ( δ a i , λ i − θ a i , λ i ) p − ε v = ∫ Ω δ a i , λ i p − ε v + O ∫ Ω δ a i , λ i 4 n − 2 − ε θ a i , λ i ∣ v ∣ .

The last integral can be deduced following the proof of Lemma 6.5 of [7] by using (15) and we obtain

(28) ∫ Ω δ a , λ 4 n − 2 − ε θ a , λ ∣ v ∣ ≤ c ∫ Ω δ a , λ 4 ∕ ( n − 2 ) θ a , λ ∣ v ∣ ≤ c ‖ v ‖ T 2 ( λ ) .

For the other one, using (15), and the fact that v ⊥ π δ a i , λ i , we obtain

(29) ∫ Ω δ a i , λ i p − ε v = c 0 − ε λ i − ε n − 2 2 ∫ Ω δ a i , λ i p v + O ε ∫ Ω δ a i , λ i p ln ( 1 + λ i 2 ∣ x − a i ∣ 2 ) ∣ v ∣ = O ε ‖ v ‖ ∫ Ω δ a i , λ i 2 n n − 2 ln 2 n n + 2 ( 1 + λ i 2 ∣ x − a i ∣ 2 ) n + 2 2 n = O ( ε ‖ v ‖ ) .

Thus, (28) and (29) imply that

(30) ∫ Ω ( π δ a i , λ i ) p − ε v = O ( ε ‖ v ‖ + ‖ v ‖ T 2 ( λ i ) ) .

Substituting (30), (26), and (25) into (22), the result follows.□

Combining (19), Lemma 2.1, and the fact that Q ε is a positive definite quadratic form, we deduce that

Proposition 2.2

Let ( α , a , λ ) ∈ O ( N , μ ) and u ̲ ≔ ∑ i = 1 N α i π δ a i , λ i . Then, for ε small, there exists a unique v ¯ ∈ E a , λ ⊥ satisfying:

(31) ⟨ ∇ I V , ε ( u ̲ + v ¯ ) , h ⟩ = 0 ∀ h ∈ E a , λ ⊥ .

Moreover, v ¯ satisfies

∥ v ¯ ∥ ≤ c R v ( a , λ ) , w i t h R v ( a , λ ) ≔ ε + ∑ i ≠ j T 3 ( ε i j ) + ∑ 1 ≤ i ≤ N T 2 ( λ i ) ,

where T 2 and T 3 are defined in Lemma 2.1.

2.2 Expansion of the gradient of the associated functional

In this section, we will provide asymptotic expansions of the gradient of functional I V , ε . We begin with the expansion with respect to the gluing parameter α ′ s , specifically proving:

Proposition 2.3

Let ( α , a , λ ) ∈ O ( N , μ ) , v ∈ E a , λ ⊥ , and u = ∑ i = 1 N α i π δ a i , λ i + v . Then, for ε small and i ∈ { 1 , … , N } , we have:

⟨ ∇ I V , ε ( u ) , π δ a i , λ i ⟩ = α i S n ( 1 − α i p − 1 − ε λ i − ε ( n − 2 ) ∕ 2 ) + O ( R α i ) ,

where

S n ≔ ∫ R n δ 0,1 2 n ∕ ( n − 2 ) , R α i ≔ ε + T 2 ( λ i ) + ∥ v ∥ 2 + ∑ i ≠ j ε i j

and T 2 is defined in Lemma 2.1.

Proof

Observe that, since v ∈ E a , λ ⊥ , it holds

(32) ⟨ ∇ I V , ε , π δ a i , λ i ⟩ = ∑ j = 1 N α j ⟨ π δ a j , λ j , π δ a i , λ i ⟩ − ∫ Ω ∣ u ∣ p − 1 − ε u π δ a i , λ i .

First, for j ≠ i , we have

(33) ⟨ π δ a j , λ j , π δ a i , λ i ⟩ = ∫ Ω ∇ ( π δ a j , λ j ) ∇ ( π δ a i , λ i ) + ∫ Ω V ( π δ a j , λ j ) ( π δ a i , λ i ) = ∫ Ω ( − Δ + V ) ( π δ a j , λ j ) ( π δ a i , λ i ) ≤ ∫ Ω δ a j , λ j n + 2 n − 2 δ a i , λ i ≤ c ε i j ,

using Estimate (E1) of [13]. Second, for j = i , as in the previous computation, we have

∥ π δ a i , λ i ∥ 2 = ∫ Ω δ a i , λ i n + 2 n − 2 ( π δ a i , λ i ) = ∫ Ω δ a i , λ i 2 n n − 2 − ∫ Ω δ a i , λ i n + 2 n − 2 θ a i , λ i .

The estimate of the first integral is well known and we have

∫ Ω δ a i , λ i 2 n n − 2 = S n + O 1 λ i n .

For the second integral, let B i ≔ B ( a i , d 0 ) , using (10) and (11), easy computations imply that

(34) ∫ Ω δ a i , λ i n + 2 n − 2 θ a i , λ i ≤ c ∫ B i R 1 ( x , a i , λ i ) δ a i , λ i 2 n n − 2 + ∫ Ω \ B i δ a i , λ i 2 n n − 2 ≤ c T 2 ( λ i ) + c λ i n ≤ c T 2 ( λ i ) .

Thus, we deduce that

(35) ∥ π δ a i , λ i ∥ 2 = S n + O ( λ − 2 , if n ≥ 5 ; λ − 2 ln λ , if n = 4 ; λ − 1 , if n = 3 ) .

Now, we focus on the last term in (32). Observe that, for b 1 , b 2 , z ∈ R such that ∣ z ∣ ≤ β ∣ b 1 ∣ for some positive constant β and γ > 0 , it holds

(36) ∣ b 1 + b 2 ∣ γ − 1 ( b 1 + b 2 ) z = ∣ b 1 ∣ γ − 1 b 1 z + γ ∣ b 1 ∣ γ − 1 b 2 z + O ( ∣ b 1 ∣ γ − 1 b 2 2 + ∣ b 2 ∣ γ + 1 ) ,

(37) ∣ b 1 + b 2 ∣ γ = ∣ b 1 ∣ γ + O ( ∣ b 1 ∣ γ − 1 ∣ b 2 ∣ + ∣ b 2 ∣ γ ) .

Thus, let u ̲ ≔ ∑ j = 1 N α j ( π δ a j , λ j ) ; using (36), we obtain

(38) ∫ Ω ∣ u ∣ p − ε − 1 u ( π δ a i , λ i ) = ∫ Ω ( u ̲ ) p − ε ( π δ a i , λ i ) + ( p − ε ) ∫ Ω ( u ̲ ) p − ε − 1 v ( π δ a i , λ i ) + O ( ∥ v ∥ 2 ) .

The first integral in the right-hand side of (38) can be written as (using (37), (27), and (9))

∫ Ω ( u ̲ ) p − ε ( π δ a i , λ i ) = α i p − ε ∫ Ω ( π δ a i , λ i ) p + 1 − ε + ∑ j ≠ i O ∫ Ω δ a i , λ i p − ε δ a j , λ j + ∫ Ω δ a j , λ j p − ε δ a i , λ i = α i p − ε ∫ Ω δ a i , λ i p + 1 − ε + O ∫ Ω δ a i , λ i p − ε θ a i , λ i + ∑ j ≠ i ∫ Ω δ a i , λ i p δ a j , λ j + ∫ Ω δ a j , λ j p δ a i , λ i = α i p − ε c 0 − ε λ i − ε ( n − 2 ) ∕ 2 S n + O ε + 1 λ i n + T 2 ( λ i ) + ∑ i ≠ j ε i j ,

where we have used ( 64 ) of [15], (34), and the last inequality in (33). Concerning the second integral in (38), it is computed in Lemma A.2.

Combining the previous estimates, the proof follows.□

We will now present a balancing formula that involves the rate of concentration λ i and the self-interaction of the bubbles ε i j . Specifically, we prove:

Proposition 2.4

Let n ≥ 4 , ( α , a , λ ) ∈ O ( N , μ ) , v ∈ E a , λ ⊥ , and u = ∑ j = 1 N α j π δ a j , λ j + v . For ε small and i ∈ { 1 , … , N } , we have

∇ I V , ε ( u ) , λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∑ j ≠ i α j c ¯ 2 λ i ∂ ε i j ∂ λ i ( 1 − α i p − 1 − ε λ i − ε ( n − 2 ) ∕ 2 − α j p − 1 − ε λ j − ε ( n − 2 ) ∕ 2 ) + c 0 − ε λ i − ε n − 2 2 c 2 α i p − ε ε − c ( n ) α i ln σ n ( λ i ) λ i 2 V ( a i ) 2 α i p − ε − 1 c 0 − ε λ i − ε n − 2 2 − 1 + O ( R λ i ) ,

where

c ( 4 ) ≔ c 0 2 mes ( S 3 ) , c ( n ) ≔ n − 2 2 c 0 2 ∫ R n ∣ x ∣ 2 − 1 ( 1 + ∣ x ∣ 2 ) n − 1 d x > 0 i f n ≥ 5 , c 2 ≔ n − 2 2 2 c 0 2 n n − 2 ∫ R n ( ∣ x ∣ 2 − 1 ) ( 1 + ∣ x ∣ 2 ) n + 1 ln ( 1 + ∣ x ∣ 2 ) d x > 0 , c ¯ 2 ≔ c 0 2 n n − 2 ∫ R n 1 ( 1 + ∣ x ∣ 2 ) ( n + 2 ) ∕ 2 d x , R λ i ≔ ε 2 + ∥ v ∥ 2 + 1 λ i n − 2 + 1 λ i 4 + ( i f n = 6 ) ln λ i λ i 4 + ∑ k ≠ j ε k j n n − 2 ln ( ε k j − 1 ) + ∑ j ≠ i 1 ( λ j λ i ) ( n − 2 ) ∕ 2 + ε i j Ξ i j + 1 λ j 3 ∕ 2 + 1 λ i 3 ∕ 2

and where Ξ i j is defined in Lemma A.3.

Proof

We will follow the proof of Proposition 2.3, and we need to be more precise in some integrals. Since v ∈ E a , λ ⊥ , we have

u , λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∑ α j π δ a j , λ j , λ i ∂ ( π δ a i , λ i ) ∂ λ i .

First, for j = i , using Lemma 6.3 of [7], we obtain

(39) π δ a i , λ i , λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∫ Ω δ a i , λ i n + 2 n − 2 λ i ∂ δ a i , λ i ∂ λ i − λ i ∂ θ a i , λ i ∂ λ i = c ( n ) ln σ n λ i λ i 2 V ( a i ) + O 1 λ i n − 2 + 1 λ i 4 + ( if n = 6 ) ln λ i λ i 4 .

Second, for j ≠ i , we have

(40) π δ a j , λ j , λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∫ Ω δ a j , λ j n + 2 n − 2 λ i ∂ δ a i , λ i ∂ λ i − λ i ∂ θ a i , λ i ∂ λ i .

Observe that, using Estimate F16 of [13] and the fact that d ( a k , ∂ Ω ) ≥ c > 0 for k ∈ { i , j } , we deduce that

∫ Ω δ a j , λ j n + 2 n − 2 λ i ∂ δ a i , λ i ∂ λ i = c ¯ 2 λ i ∂ ε i j ∂ λ i + O ε i j n n − 2 ln ( ε i j − 1 ) + 1 λ j ( n + 2 ) ∕ 2 λ i ( n − 2 ) ∕ 2 .

Furthermore, using Lemma A.3, equation (40) becomes

(41) π δ a j , λ j , λ i ∂ ( π δ a i , λ i ) ∂ λ i = c ¯ 2 λ i ∂ ε i j ∂ λ i + O ε i j n n − 2 ln ( ε i j − 1 ) + 1 ( λ j λ i ) ( n − 2 ) ∕ 2 + Ξ i j ε i j .

Thus, we derive that

(42) u , λ i ∂ ( π δ a i , λ i ) ∂ λ i = α i c ( n ) ln σ n λ i λ i 2 V ( a i ) + c ¯ 2 ∑ j ≠ i α j λ i ∂ ε i j ∂ λ i + O 1 λ i n − 2 + 1 λ i 4 + ( if n = 6 ) ln λ i λ i 4 + ∑ j ≠ i O ε i j n n − 2 ln ( ε i j − 1 ) + 1 ( λ j λ i ) ( n − 2 ) ∕ 2 + ε i j Ξ i j .

Now, using (36), we obtain

(43) ∫ Ω ∣ u ∣ p − ε − 1 u λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∫ Ω ( u ̲ ) p − ε λ i ∂ ( π δ a i , λ i ) ∂ λ i + ( p − ε ) ∫ Ω ( u ̲ ) p − ε − 1 v λ i ∂ ( π δ a i , λ i ) ∂ λ i + O ( ‖ v ‖ 2 ) .

The last integral is computed in Lemma A.2. For the first one, we need the following formula. Let 1 < γ ≤ 3 , b j > 0 for j ∈ { 1 , … , N } , and s be such that ∣ s ∣ ≤ c b i for some index i . Then, it holds

(44) ∑ j = 1 N b j γ s = ∑ j = 1 N b j γ s + γ b i γ − 1 ∑ j ≠ i b j s + ∑ k ≠ j O ( ( b k b j ) ( γ + 1 ) ∕ 2 ) .

Thus, we obtain

(45) ∫ Ω ( u ̲ ) p − ε λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∑ 1 ≤ j ≤ N α j p − ε ∫ Ω ( π δ a j , λ j ) p − ε λ i ∂ ( π δ a i , λ i ) ∂ λ i + ( p − ε ) ∑ j ≠ i ∫ Ω ( α i π δ a i , λ i ) p − ε − 1 ( α j π δ a j , λ j ) λ i ∂ ( π δ a i , λ i ) ∂ λ i + ∑ j ≠ k O ∫ Ω ( δ a j , λ j δ a k , λ k ) n n − 2 .

First, we remark that the remainder term is computed in (24). Second, the first integral, with j = i , is computed in [7] (see equation (45)) and we have

(46) ∫ Ω ( π δ a i , λ i ) p − ε λ i ∂ ( π δ a i , λ i ) ∂ λ i = c 0 − ε λ i − ε n − 2 2 − c 2 ε + 2 c ( n ) ln σ n λ i λ i 2 + O ε 2 + 1 λ i n − 2 + 1 λ i 4 + ( if n = 6 ) ln λ i λ i 4 .

Third, we focus on estimating the first integral of (45) with j ≠ i . Using (27), (9), and (15), we deduce that

∫ Ω ( π δ a j , λ j ) p − ε λ i ∂ ( π δ a i , λ i ) ∂ λ i = ∫ Ω ( δ a j , λ j p − ε + O ( δ a j , λ j p − 1 θ a j , λ j ) ) λ i ∂ π δ a i , λ i ∂ λ i = c 0 − ε λ j − ε ( n − 2 ) ∕ 2 ∫ Ω δ a j , λ j p λ i ∂ π δ a i , λ i ∂ λ i + O ∫ Ω θ a j , λ j δ a j , λ j p − 1 δ a i , λ i + ε ∫ Ω δ a j , λ j p ln ( 1 + λ j 2 ∣ x − a j ∣ 2 ) δ a i , λ i .

Observe that, let B j ≔ B ( a j , d 0 ) , we have

(47) ∫ Ω δ a j , λ j p − 1 θ a j , λ j δ a i , λ i ≤ c ∫ B j R 1 ( x , a j , λ j ) δ a j , λ j p δ a i , λ i + c λ j ( n + 2 ) ∕ 2 ∫ δ a i , λ i ≤ c ln σ n λ j λ j 2 ∫ B j δ a j , λ j p δ a i , λ i + ∫ B j ∣ x − a j ∣ 2 ∣ ln ∣ x − a j ∣ ∣ σ n δ a j , λ j p δ a i , λ i + c λ j ( n + 2 ) ∕ 2 ∫ δ a i , λ i .

Using Holder’s inequality and Lemma 2.2 of [14], we deduce that

∫ B j ∣ x − a j ∣ 2 ∣ ln σ n ∣ x − a j ∣ ∣ δ a j , λ j p δ a i , λ i ≤ c λ j 3 ∕ 2 ∫ B j ∣ ln ∣ x − a j ∣ ∣ σ n ∣ x − a j ∣ δ a j , λ j n − 1 n − 2 δ a i , λ i ≤ c λ j 3 ∕ 2 ∫ B j ∣ ln ∣ x − a j ∣ ∣ ( n − 1 ) σ n ∣ x − a j ∣ n − 1 1 ∕ ( n − 1 ) ∫ B j δ a j , λ j n − 1 n − 2 2 δ a i , λ i n − 1 n − 2 ( n − 2 ) ∕ ( n − 1 ) ≤ c λ j 3 ∕ 2 ε i j .

Thus, (47) becomes

(48) ∫ Ω δ a j , λ j p − 1 θ a j , λ j δ a i , λ i ≤ c λ j 3 ∕ 2 ε i j + c λ i ( n − 2 ) ∕ 2 λ j ( n + 2 ) ∕ 2 .

Hence, we obtain

∫ Ω ( π δ a j , λ j ) p − ε λ i ∂ ( π δ a i , λ i ) ∂ λ i = c 0 − ε λ j − ε n − 2 2 π δ a j , λ j , λ i ∂ ( π δ a i , λ i ) ∂ λ i + O 1 λ i ( n − 2 ) ∕ 2 λ j ( n + 2 ) ∕ 2 + ε i j ε + 1 λ j 3 ∕ 2 .

In the same way, we deduce the estimate of the second integral of (45), and we obtain

p ∫ Ω ( π δ a i , λ i ) p − ε − 1 λ i ∂ ( π δ a i , λ i ) ∂ λ i π δ a j , λ j = c 0 − ε λ i − ε n − 2 2 π δ a j , λ j , λ i ∂ ( π δ a i , λ i ) ∂ λ i + O 1 λ j ( n − 2 ) ∕ 2 λ i ( n + 2 ) ∕ 2 + ε i j ε + 1 λ i 3 ∕ 2 .

Finally, combining (42), (43), (45), (41), (46), and Lemma A.2, the result follows.□

Next, we present a balancing condition that involves the concentration points.

Proposition 2.5

Let n ≥ 4 , ( α , a , λ ) ∈ O ( N , μ ) , v ∈ E a , λ ⊥ , and u = ∑ j = 1 N α j ( π δ a j , λ j ) + v . For ε small and i ∈ { 1 , … , N } , we have

∇ I V , ε ( u ) , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = c 2 ( n ) α i ln σ n ( λ i ) λ i 3 ∇ V ( a i ) 2 α i p − ε − 1 c 0 − ε λ i − ε n − 2 2 − 1 + c ¯ 2 ∑ j ≠ i α j 1 λ i ∂ ε i j ∂ a i 1 − c 0 − ε λ i − ε n − 2 2 α i p − ε − 1 − c 0 − ε λ i − ε n − 2 2 α j p − ε − 1 + O ( R a i ) ,

with

R a i ≔ ∑ j ≠ i λ j ∣ a i − a j ∣ ε i j n + 1 n − 2 + ∥ v ∥ 2 + 1 λ i 4 + ( i f n = 5 ) ln λ i λ i 4 + ∑ 1 λ k n − 1 + ∑ k ≠ j ε k j n ∕ ( n − 2 ) ln ε k j − 1 + ∑ ε i j ε + 1 λ i + 1 λ j 3 ∕ 2 + ε ln σ n λ i λ i 2 ,

where

c 2 ( 4 ) ≔ 1 2 c 0 2 meas ( S 3 ) a n d c 2 ( n ) ≔ n − 2 n c 0 2 ∫ R n ∣ x ∣ 2 ( 1 + ∣ x ∣ 2 ) n − 1 , i f n ≥ 5 .

Proof

We will follow the proof of Proposition 2.4. Since v ∈ E a , λ ⊥ , we have

u , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = ∑ α j π δ a j , λ j , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i .

First, the scalar product, with j = i , is computed in [7] and we have

(49) π δ a i , λ i , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = − c 2 ( n ) ln σ n λ i λ i 3 ∇ V ( a i ) + O 1 λ i n − 1 + 1 λ i 4 + ( if n = 5 ) ln λ i λ i 4 .

Second, for j ≠ i , using (11) and (12), the last inequality in (33) and (F11) of [13], we obtain

(50) π δ a j , λ j , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = ∫ Ω δ a j , λ j n + 2 n − 2 1 λ i ∂ δ a i , λ i ∂ a i − 1 λ i ∂ θ a i , λ i ∂ a i = ∫ R n δ a j , λ j n + 2 n − 2 1 λ i ∂ δ a i , λ i ∂ a i + O 1 λ j ( n + 2 ) ∕ 2 λ i n ∕ 2 + 1 λ i ∫ Ω δ a j , λ j n + 2 n − 2 δ a i , λ i = c ¯ 2 1 λ i ∂ ε i j ∂ a i + O λ j ∣ a i − a j ∣ ε i j n + 1 n − 2 + 1 λ i ε i j + 1 λ i n ∕ 2 λ j ( n + 2 ) ∕ 2 .

Thus, we derive that

(51) u , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = − α i c 2 ( n ) ln σ n λ i λ i 3 ∇ V ( a i ) + c ¯ 2 α j 1 λ i ∂ ε i j ∂ a i + O ∑ j ≠ i λ j ∣ a i − a j ∣ ε i j n + 1 n − 2 + 1 λ i ε i j + 1 λ i n ∕ 2 λ j ( n + 2 ) ∕ 2 + 1 λ i n − 1 + 1 λ i 4 + ( if n = 5 ) ln λ i λ i 4 .

Now, note that (43) and (45) hold true with λ i − 1 ∂ ( π δ a i , λ i ) ∕ ∂ a i instead of λ i ∂ ( π δ a i , λ i ) ∕ ∂ λ i . Furthermore, equation (53) of [7] implies that

(52) ∫ Ω ( π δ a i , λ i ) p − ε 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = − 2 c 2 ( n ) c 0 − ε λ i − ε n − 2 2 ln σ n λ i λ i 3 ∇ V ( a i ) + O 1 λ i n − 1 + 1 λ i 4 + ( if n = 5 ) ln λ i λ i 4 + ε ln σ n λ i λ i 2 .

In addition, for j ≠ i , using (27), (9), (48), and (15), we deduce that

∫ Ω ( π δ a j , λ j ) p − ε 1 λ i ∂ ( π δ a i , λ i ) ∂ a i = ∫ Ω ( δ a j , λ j p − ε + O ( δ a j , λ j p − 1 θ a j , λ j ) ) 1 λ i ∂ π δ a i , λ i ∂ a i = c 0 − ε λ j − ε n − 2 2 ∫ Ω δ a j , λ j p 1 λ i ∂ π δ a i , λ i ∂ a i + O ∫ Ω θ a j , λ j δ a j , λ j p − 1 δ a i , λ i + ε ∫ Ω δ a j , λ j p ln ( 1 + λ j 2 ∣ x − a j ∣ 2 ) δ a i , λ i = c 0 − ε λ j − ε n − 2 2 π δ a j , λ j , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i + O 1 λ i ( n − 2 ) ∕ 2 λ j ( n + 2 ) ∕ 2 + ε i j ε + 1 λ j 3 ∕ 2 .

In the same way, we obtain

p ∫ Ω ( π δ a i , λ i ) p − ε − 1 1 λ i ∂ ( π δ a i , λ i ) ∂ a i π δ a j , λ j = c 0 − ε λ i − ε n − 2 2 π δ a j , λ j , 1 λ i ∂ ( π δ a i , λ i ) ∂ a i + O 1 λ j ( n − 2 ) ∕ 2 λ i ( n + 2 ) ∕ 2 + ε i j ε + 1 λ i 3 ∕ 2 .

Combining the previous estimates, the result follows.□

3 Interior blowing-up solutions with clustered bubbles

This section is devoted to the proof of Theorems 1.1 and 1.3. We start by proving Theorem 1.1. The proof structure adopts a method comparable to the one presented in [15–17]. Let n ≥ 4 , b ∈ Ω be a non-degenerate critical point of V and ( z ¯ 1 , … , z ¯ N ) be a non-degenerate critical point of F b , N , where F b , N is defined by (2). We start by introducing a parameterization of the neighborhood of the desired constructed solutions. Let

(53) O 1 ( N , b , ε ) = { ( α , λ , a , v ) ∈ ( R + ) N × ( R + ) N × Ω N × H 1 ( Ω ) : v ∈ E a , λ ⊥ , ‖ v ‖ < ε , ∣ α i − 1 ∣ < ε ln 2 ε , 1 c < ln σ n ( λ i ) λ i 2 ε < c , ∣ a i − b − η ( ε ) σ z ¯ i ∣ ≤ η ( ε ) η 0 ∀ 1 ≤ i ≤ N ,

where η 0 is a small constant, σ 4 = 1 , σ n = 0 for n ≥ 5 , E a , λ is defined by (17),

(54) η ( ε ) ≔ ε γ if n ≥ 5 ∣ ln ε ∣ − 1 ∕ 4 if n = 4 , γ = n − 4 2 n and σ = c ¯ 2 c 2 ( n ) 1 ∕ n c 2 c ( n ) V ( b ) γ

with c 2 , c ¯ 2 , c 2 ( n ) , and c ( n ) are the constants defined in Propositions 2.4 and 2.5.

In addition, we consider the following function:

ψ ε : O 1 ( N , b , ε ) → R , ( α , λ , a , v ) ↦ ψ ε ( α , λ , a , v ) = I V , ε ∑ i = 1 N α i π δ a i , λ i + v .

We note that ( α , λ , a , v ) is a critical point of ψ ε if and only if u = ∑ i = 1 N α i π δ a i , λ i + v is a critical point of I V , ε . Thus, we need to look for critical points of ψ ε . Since the variable v belongs to E a , λ ⊥ , the Lagrange multiplier theorem allows us to obtain the following proposition.

Proposition 3.1

( α , λ , a , v ) ∈ O 1 ( N , b , ε ) is a critical point of ψ ε if and only if there exists ( A , B , C ) ∈ R N × R N × ( R n ) N such that the following holds:

(55) ∂ ψ ε ∂ α i ( α , λ , a , v ) = 0 ∀ i ∈ { 1 , … , N } ,

(56) ∂ ψ ε ∂ λ i ( α , λ , a , v ) = B i ⟨ v , λ i ∂ 2 ( π δ a i , λ i ) ∂ λ i 2 ⟩ + ∑ j = 1 n C i j v , 1 λ i ∂ 2 ( π δ a i , λ i ) ∂ λ i ∂ a i j ∀ i ∈ { 1 , … , N } ,

(57) ∂ ψ ε ∂ a i ( α , λ , a , v ) = B i ⟨ v , λ i ∂ 2 ( π δ a i , λ i ) ∂ λ i ∂ a i ⟩ + ∑ j = 1 n C i j v , 1 λ i ∂ 2 ( π δ a i , λ i ) ∂ a i j ∂ a i ∀ i ∈ { 1 , … , N } ,

(58) ∂ ψ ε ∂ v ( α , λ , a , v ) = ∑ k = 1 N A k π δ a k , λ k + B k λ k ∂ ( π δ a k , λ k ) ∂ λ k + ∑ j = 1 n C k j 1 λ k ∂ ( π δ a k , λ k ) ∂ a k j .

To prove Theorem 1.1, we will make a careful study of the previous equations. Note that

(59) ∂ ψ ε ∂ α i ( α , λ , a , v ) = ∇ I V , ε ∑ i = 1 N α i π δ a i , λ i + v , π δ a i , λ i ,

(60) ∂ ψ ε ∂ λ i ( α , λ , a , v ) = ∇ I V , ε ∑ i = 1 N α i π δ a i , λ i + v , α i ∂ ( π δ a i , λ i ) ∂ λ i ,

(61) ∂ ψ ε ∂ a i ( α , λ , a , v ) = ∇ I V , ε ∑ i = 1 N α i π δ a i , λ i + v , α i ∂ ( π δ a i , λ i ) ∂ a i ,

(62) ∂ ψ ε ∂ v ( α , λ , a , v ) = ∇ I V , ε ∑ i = 1 N α i π δ a i , λ i + v .

Let ( α , λ , a , 0 ) ∈ O 1 ( N , b , ε ) , which is defined by (53). We are going to solve the system defined by equations (55)–(58). The strategy of the proof of Theorem 1.1 is to reduce the problem to a finite-dimensional system. Proposition 2.2 allows us to obtain such a reduction by finding v ¯ verifying equation (58). Combining equations (55), …, (62), we see that u = ∑ i = 1 N α i π δ a i , λ i + v ¯ is a critical point of I V , ε if and only if ( α , λ , a ) solves the following system: for each 1 ≤ i ≤ N ,

(63) ( E α i ) ⟨ ∇ I V , ε ( u ) , π δ a i , λ i ⟩ = 0 ,

(64) ( E λ i ) ∇ I V , ε ( u ) , α i ∂ ( π δ a i , λ i ) ∂ λ i = B i v ¯ , λ i ∂ 2 ( π δ a i , λ i ) ∂ λ i 2 + ∑ j = 1 n C i j v ¯ , 1 λ i ∂ 2 ( π δ a i , λ i ) ∂ λ i ∂ a i j ,

(65) ( E a i ) ∇ I V , ε ( u ) , α i ∂ ( π δ a i , λ i ) ∂ a i = B i v ¯ , λ i ∂ 2 ( π δ a i , λ i ) ∂ λ i ∂ a i + ∑ j = 1 n C i j v ¯ , 1 λ i ∂ 2 ( π δ a i , λ i ) ∂ a i j ∂ a i .

We are looking for ( α , λ , a ) ∈ O 2 ( N , b , ε ) solution of the system defined by equations (63), (64), and (65), where O 2 ( N , b , ε ) is defined by

(66) O 2 ( N , b , ε ) = { ( α , λ , a ) ∈ ( R + ) N × ( R + ) N × Ω N : ( α , λ , a , 0 ) ∈ O 1 ( N , b , ε ) } .

In order to work with a simpler system, we make the following change of variables, for 1 ≤ i ≤ N ,

(67) β i = 1 − α i p − 1 , ( a i − b ) = η ( ε ) σ ( ζ i + z ¯ i ) , 1 λ i 2 = c 2 c ( n ) 1 V ( b ) ε ( ∣ ln ε ∣ ∕ 2 ) − σ n ( 1 + ∧ i ) .

Using these changes of variables, it is easy to see that

(68) c ≤ λ i ∣ a i − a j ∣ ε 2 ∕ n ∣ ln ε ∣ − σ n ∕ 4 ≤ c ′ ,

(69) ε i j = 1 ( λ i λ j ∣ a i − a j ∣ 2 ) n − 2 2 1 + O 1 λ i 2 ∣ a i − a j ∣ 2 + 1 λ j 2 ∣ a i − a j ∣ 2 = 1 ( λ i λ j ∣ a i − a j ∣ 2 ) n − 2 2 ( 1 + O ( ε 4 ∕ n ∣ ln ε ∣ − σ n ∕ 2 ) ) = O ( ε 2 γ + 1 ∣ ln ε ∣ − σ n ∕ 2 ) ,

(70) ∂ ε i j ∂ a i = ( n − 2 ) λ i λ j ( a j − a i ) ε i j n ∕ ( n − 2 ) = ( n − 2 ) ( a j − a i ) ( λ i λ j ) ( n − 2 ) ∕ 2 ∣ a i − a j ∣ n ( 1 + O ( ε 4 ∕ n ∣ ln ε ∣ − σ n ∕ 2 ) ) = ( n − 2 ) ( ε ∣ ln ε ∣ − σ n ) ( n − 2 ) ∕ 2 σ n − 1 η ( ε ) n − 1 c 2 c ( n ) V ( b ) n − 2 2 ( ζ j + z ¯ j − ζ i − z ¯ i ) ∣ ζ i + z ¯ i − ζ j − z ¯ j ∣ n × ( 1 + R ( ∧ i , ∧ j ) ) ( 1 + O ( ε 4 ∕ n ∣ ln ε ∣ − σ n ∕ 2 ) )

(71) = O ( ε γ + 1 ∣ ln ε ∣ − σ n ∕ 4 ) ,

where

(72) R ( ∧ i , ∧ j ) = n − 2 4 ∧ i + n − 2 4 ∧ j + O ( ∧ i 2 ) + O ( ∧ j 2 ) .

Next, using Propositions 2.1–2.5, we deduce that the following estimates hold:

Lemma 3.2

For ε small, the following statements hold:

‖ v ¯ ‖ ≤ c ε , R α i ≤ c ε , R λ i ≤ c ε 2 γ + 1 ∣ ln ε ∣ − σ n ∕ 2 , R a i ≤ c ε 2 ∣ ln ε ∣ + ε ( 5 ∕ 2 ) − ( 4 ∕ n ) , i f n ≥ 5 , ε 3 ∕ 2 ∣ ln ε ∣ − 1 , i f n = 4 .

where R α i , R λ i , and R a i are defined in Propositions 2.3–2.5, respectively.

Now, arguing as in the proof of Lemma 4.2 of [7] and using Lemma A.4, we derive that the constants A i ’s, B i ’s and C i j ’s which appear in equations ( E v ) , ( E λ i ) , and ( E a i ) satisfy the following estimates:

Lemma 3.3

Let ( α , λ , a ) ∈ O 2 ( N , b , ε ) . Then, for ε small, the following statements hold:

A i = O ( ε ln 2 ε ) , B i = O ( ε ) a n d ∣ C i j ∣ ≤ c ε γ + 3 ∕ 2 , i f n ≥ 5 , ε 3 ∕ 2 ∣ ln ε ∣ − 3 ∕ 4 , i f n = 4 , ∀ i ≤ N , ∀ j ≤ n .

Next, our aim is to rewrite equations ( E α i ) , ( E λ i ) , and ( E a i ) in a simple form.

Lemma 3.4

For ε small, equations ( E α i ) , ( E λ i ) , and ( E a i ) are equivalent to the following system:

( S ′ ) β i = O ( ε ∣ ln ε ∣ ) ∀ 1 ≤ i ≤ N , ( F α i ′ ) ∧ i = O ( ∣ β i ∣ + ε 2 γ ∣ ln ε ∣ − σ n ∕ 2 ) ∀ 1 ≤ i ≤ N , ( F λ i ′ ) D 2 V ( b ) ( ζ i , ⋅ ) − ( n − 2 ) ∑ j ≠ i ζ j − ζ i ∣ z ¯ j − z ¯ i ∣ n + n ( n − 2 ) ∑ j ≠ i z ¯ j − z ¯ i ∣ z ¯ j − z ¯ i ∣ 2 , ζ j − ζ i z ¯ j − z ¯ i ∣ z ¯ j − z ¯ i ∣ n − ( n − 2 ) ∑ j ≠ i z ¯ j − z ¯ i ∣ z ¯ j − z ¯ i ∣ n n − 6 4 ∧ i + n − 2 4 ∧ j = O ( R ′ S ) ∀ 1 ≤ i ≤ N , ( F a i ′ )

where

R S ′ = ∑ j = 1 N ∧ j 2 + ∑ j = 1 N ∣ ζ j ∣ 2 + η ( ε ) + ε 1 2 − γ ∣ ln ε ∣ .

Proof

First, using Proposition 2.3, Lemma 3.2, and the fact that ( α , λ , a ) ∈ O 2 ( N , b , ε ) , we see that ( E α i ) is equivalent to the equation ( F α i ′ ) . Second, using Lemmas 3.3 and A.4, we write

∇ I ε ( u ) , λ i ∂ π δ a i , λ i ∂ λ i = O ∣ B i ∣ ‖ v ‖ λ i 2 ∂ 2 π δ a i , λ i ∂ λ i 2 + ∑ j = 1 N ∣ C i j ∣ ‖ v ‖ ∂ 2 π δ a i , λ i ∂ λ i ∂ a i j = O ( ε ‖ v ‖ ) .

Using Proposition 2.4, Lemma 3.2 and (69), we obtain

c 2 ε − c ( n ) ln σ n λ i λ i 2 V ( a i ) = O ( ε ∣ β i ∣ + ε 2 γ + 1 ∣ ln ε ∣ − σ n ∕ 2 ) .

But we have

V ( a i ) = V ( b ) + O ( ∣ a i − b ∣ 2 ) = V ( b ) + O ( η ( ε ) 2 ) .

This implies that ( E λ i ) is equivalent to the equation ( F λ i ′ ) .

To deal with the third equation ( E a i ) , using Lemma A.4, we write

(73) ∇ I ε ( u ) , 1 λ i ∂ π δ a i , λ i ∂ a i = O ∣ B i ∣ ‖ v ‖ ∂ 2 π δ a i , λ i ∂ λ i ∂ a i + ∑ j = 1 N ∣ C i j ∣ ‖ v ‖ 1 λ i 2 ∂ 2 π δ a i , λ i ∂ a i ∂ a i j = O ( ε ‖ v ‖ ) .

But, combining (70), Proposition 2.5 and Lemma 3.2, (73) becomes

(74) c 2 ( n ) ln σ n λ i λ i 3 ∇ V ( a i ) − ( n − 2 ) c ¯ 10 ε η ( ε ) λ i ∑ j ≠ i ( ζ j + z ¯ j − ζ i − z ¯ i ) ∣ ζ j + z ¯ j − ζ i − z ¯ i ∣ n 1 + n − 2 4 ∧ i + n − 2 4 ∧ j = O ε 2 ∣ ln ε ∣ σ n ∕ 2 + R a i + η ( ε ) ε 3 ∕ 2 ∣ ln ε ∣ − σ n ∕ 2 ∑ ( ∣ β j ∣ + ∧ j 2 ) ,

where

c ¯ 10 = c ¯ 2 σ n − 1 c 2 c ( n ) V ( b ) ( n − 2 ) ∕ 2 .

Observe that

(75) ∇ V ( a i ) = D 2 V ( b ) ( a i − b , ⋅ ) + O ( ∣ a i − b ∣ 2 ) = η ( ε ) σ D 2 V ( b ) ( ζ i + z ¯ i , ⋅ ) + O ( η ( ε ) 2 ) .

Combining (74), (75), and (54), we obtain

(76) D 2 V ( b ) ( ζ i + z ¯ i , ⋅ ) − ( n − 2 ) ∑ j ≠ i ( ζ j + z ¯ j − ζ i − z ¯ i ) ∣ ζ j + z ¯ j − ζ i − z ¯ i ∣ n 1 + n − 6 4 ∧ i + n − 2 4 ∧ j = O η ( ε ) + ε 1 2 − γ ∣ ln ε ∣ + ∑ ( ∣ β j ∣ + ∧ j 2 ) .

But we have

(77) D 2 V ( b ) ( ζ i + z ¯ i , ⋅ ) = D 2 V ( b ) ( ζ i , ⋅ ) + D 2 V ( b ) ( z ¯ i , ⋅ )

and

(78) ( ζ j + z ¯ j − ζ i − z ¯ i ) ∣ ζ j + z ¯ j − ζ i − z ¯ i ∣ n = ( z ¯ j − z ¯ i ) ∣ z ¯ j − z ¯ i ∣ n + ( ζ j − ζ i ) ∣ z ¯ j − z ¯ i ∣ n 1 − n ( z ¯ j − z ¯ i ) ∣ z ¯ j − z ¯ i ∣ 2 , ζ j − ζ i + O ( ∣ ζ i ∣ 2 + ∣ ζ j ∣ 2 ) = ( z ¯ j − z ¯ i ) ∣ z ¯ j − z ¯ i ∣ n − n ( z ¯ j − z ¯ i ) ∣ z ¯ j − z ¯ i ∣ 2 , ζ j − ζ i ( z ¯ j − z ¯ i ) ∣ z ¯ j − z ¯ i ∣ n + ( ζ j − ζ i ) ∣ z ¯ j − z ¯ i ∣ n + O ( ∣ ζ i ∣ 2 + ∣ ζ j ∣ 2 ) .

Combining (76)–(78) and the fact that ( z ¯ 1 , … , z ¯ N ) is a critical point of F N , b , we see that equation ( E a i ) is equivalent to the equation ( F a i ′ ) , which completes the proof of Lemma 3.4.□

Now, we are ready to prove our results related to the construction of clustered bubbling solutions.

Proof of Theorem 1.1

Note that the system ( F a 1 ′ ) , … , ( F a N ′ ) is equivalent to

1 2 D 2 F N , b ( z ¯ 1 , … , z ¯ N ) ( ζ 1 , … , ζ N ) − ( n − 2 ) ( Γ 1 , … , Γ N ) = O ∑ j = 1 N ( ∧ j 2 + ∣ ζ j ∣ 2 ) + η ( ε ) + ε 1 2 − γ ∣ ln ε ∣ ,

where

Γ i ≔ ∑ j ≠ i z ¯ j − z ¯ i ∣ z ¯ j − z ¯ i ∣ n n − 6 4 ∧ i + n − 2 4 ∧ j , ∀ 1 ≤ i ≤ N .

As in the proof of Theorem 1 of [15], we define a linear map by taking the left-hand side of the system defined by ( F α i ′ ) , ( F λ i ′ ) , and ( F a i ′ ) . Since ( z ¯ 1 , … , z ¯ N ) is a non-degenerate critical point of F N , b , we deduce that such a linear map is invertible and arguing as in the proof of Theorem 1 of [15], we derive that the system ( S ′ ) has a solution ( β ε , ∧ ε , ζ ε ) for ε small. This implies that ( P V , ε ) admits a solution u ε , b = ∑ i = 1 N α i , ε π δ a i , ε , λ i , ε + v ε and, by construction, properties (3)–(6) are satisfied. The proof of Theorem 1.1 is thereby complete.□

Proof of Theorem 1.3

Let b 1 and b 2 be two non-degenerate critical points of V . First, we observe that in the equation ( E λ i ) , the interaction between the bubbles of two different blocks is of the order of ε ( n − 2 ) ∕ 2 , which is negligible in front of the main term whose order is ε . This implies that terms of this type will fit into the rest. Second, we note that in the equation ( E a i ) , the interaction between the bubbles of two distinct blocks is of order ( ε ( n − 1 ) ∕ 2 if n ≥ 5 ; ( ε ∕ ∣ ln ε ∣ ) 3 ∕ 2 if n = 4 ), which is negligible compared to the main term of order ( ε 2 − 2 n if n ≥ 5 ; ∣ ln ε ∣ − 3 ∕ 4 ε 3 ∕ 2 if n = 4 ). Consequently, terms of this form can be incorporated into the remaining terms. Hence, arguing as in the proof of Theorem 1.1 and taking a new system ( ( S 1 ) , … , ( S m ) ) with each ( S i ) representing the system studied in the proof of Theorem 1.1, the proof of the theorem follows.□

4 Conclusion

In this article, we have investigated the existence of solutions to a nonlinear elliptic problem with Dirichlet boundary conditions, involving a slightly subcritical exponent for Sobolev embedding H 0 1 ( Ω ) ↪ L q ( Ω ) . Through a careful asymptotic analysis of the gradient of the associated Euler-Lagrange functional near the “bubbles,” we successfully constructed solutions that exhibit bubble formation clustered at interior points. The methodology adopted is tailored to variational problems. While our work demonstrates the existence of solutions with clustered bubbles for the given problem, several promising avenues for further research and open questions remain:

  1. Location of the concentration points: this article focuses on the construction of interior bubbling solutions with clustered bubbles. A natural extension of this work would be to investigate the existence of solutions that concentrate at non-isolated interior points that approach the boundary in the limit.

  2. Impact of the nature of critical points: the solutions constructed in this article are based on the assumption that the chosen critical point b of the potential V is non-degenerate, and that the corresponding function ℱ b , N has a non-degenerate critical point, where ℱ b , N is defined by (2) with N denoting the number of bubbles clustered at point b. A natural question arises: what occurs if b is degenerate, or if ℱ b , N lacks non-degenerate critical points?

  3. Impact of subcritical exponent: the present work addresses a slightly subcritical exponent for Sobolev embedding. Future investigations could extend the analysis to exponents that are slightly supercritical, i.e., when ε < 0 but close to zero.

Acknowledgment

The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).

  1. Funding information: The Article Processing Charge of this article was supported by the Deanship of Graduate Studies and Scientific Research at Qassim University.

  2. Author contributions: Both authors contributed significantly and equally to writing this article. Both authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.

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

  4. Data availability statement: All data generated or analyzed during this study are included in this published article.

Appendix

In this appendix, we compile several estimates necessary for the analysis presented in this article. We begin with the following result, which is taken from [14] (see Lemma 2.2).

Lemma A.1

Let a i , a j , λ i , and λ j be such that ε i j is small. For α and β satisfying α + β = 2 n ∕ ( n − 2 ) and α ≠ β , it holds

∫ Ω δ i α δ j β ≤ c ε i j min ( α , β ) .

We now prove some estimations involving bubbles and the approximate solutions.

Lemma A.2

For i ∈ { 1 , … , N } and j ∈ { 1 , … , n } , let

φ i ∈ π δ a i , λ i , λ i ∂ ( π δ a i , λ i ) ∂ λ i , 1 λ i ∂ ( π δ a i , λ i ) ∂ ( a i ) j .

Let u ̲ ≔ ∑ i = 1 N α i π δ a i , λ i and v ∈ E a , λ ⊥ , it holds

∫ Ω ( u ̲ ) p − ε − 1 v φ i ≤ c ‖ v ‖ ε + T 2 ( λ i ) + ∑ j ≠ i T 3 ( ε i j ) ,

where T 2 ( λ ) and T 3 ( ε i j ) are defined in Lemma 2.1.

Proof

Observe that, from (9), we deduce that ∣ φ i ∣ ≤ c π δ a i , λ i . Thus, using the fact that: for b 1 , b 2 , z ∈ R such that ∣ z ∣ ≤ c ∣ b 1 ∣ ,

(A1) ∣ ∣ b 1 + b 2 ∣ γ z − ∣ b 1 ∣ γ z ∣ ≤ c ∣ b 1 b 2 ∣ ( γ + 1 ) ∕ 2 , if γ ≤ 1 , ∣ b 1 ∣ γ ∣ b 2 ∣ + ∣ b 2 ∣ γ ∣ b 1 ∣ , if γ > 1 ,

it follows that

∫ Ω ( u ̲ ) p − ε − 1 v φ i = α i p − ε − 1 ∫ Ω ( π δ a i , λ i ) p − ε − 1 v φ i + ∑ j ≠ i O ∫ Ω ( δ a i , λ i δ a j , λ j ) p 2 ∣ v ∣ , if n ≥ 6 , O ∫ Ω δ a i , λ i p − 1 δ a j , λ j ∣ v ∣ + ∫ Ω δ a i , λ i δ a j , λ j p − 1 ∣ v ∣ , if n ≤ 5 .

Note that, for n ≤ 5 (which implies that p − 1 > 1 ), for each k and ℓ , it holds

∫ Ω δ a k , λ k p − 1 δ a ℓ , λ ℓ ∣ v ∣ ≤ c ∥ v ∥ ∫ Ω ( δ a k , λ k p − 1 δ a ℓ , λ ℓ ) 2 n n + 2 n + 2 2 n ≤ c ∥ v ∥ ε k ℓ ,

using Lemma A.1. However, using (24), we derive that

∫ Ω ( δ a i , λ i δ a j , λ j ) p 2 ∣ v ∣ ≤ c ∥ v ∥ ∫ Ω ( δ a i , λ i δ a j , λ j ) n n − 2 n + 2 2 n ≤ c ∥ v ∥ ε i j n + 2 2 ( n − 2 ) ( ln ε i j − 1 ) n + 2 2 n .

For the other integral, following the end of the proof of Lemma 6.6 of [7], we deduce that

∫ Ω ( π δ a i , λ i ) p − ε − 1 v φ i ≤ c ∥ v ∥ ( ε + T 2 ( λ i ) ) .

Hence, the proof of lemma is complete.□

Lemma A.3

Let a 1 , a 2 ∈ Ω be such that d ( a i , ∂ Ω ) ≥ c > and λ 1 , λ 2 be large reals such that ε 12 is small. Let ψ 1 ∈ { θ a 1 , λ 1 , λ 1 ∂ θ a 1 , λ 1 ∕ ∂ λ 1 } . It holds that

∫ Ω δ a 2 , λ 2 n + 2 n − 2 ∣ ψ 1 ∣ ≤ c ( λ 1 λ 2 ) ( n − 2 ) ∕ 2 + c ε 12 1 ∕ λ 1 , i f n = 3 , ln λ 1 ∕ λ 1 2 + ∣ a 1 − a 2 ∣ 2 ∣ ln ∣ a 1 − a 2 ∣ ∣ + 1 ∕ λ 2 3 ∕ 2 , i f n = 4 , 1 ∕ λ 1 2 + ∣ a 1 − a 2 ∣ 2 + 1 ∕ λ 2 3 ∕ 2 , i f n ≥ 5 . ︸ Ξ 12

Proof

First, we remark that we can modify the proof to improve the power of λ 2 . It can be β with β < 2 .

Let B 1 ≔ B ( a 1 , r 1 ) , where r 1 ≔ ( 1 ∕ 2 ) min ( d a 1 , e − 4 ) ; using (10) and (11), we obtain

∫ Ω δ a 2 , λ 2 n + 2 n − 2 ∣ ψ 1 ∣ ≤ c ∫ B 1 R a 1 , λ 1 1 ( x ) δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 + ∫ Ω \ B 1 δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ,

where R a 1 , λ 1 1 is defined in (12). For the last integral, it is easy to obtain

∫ Ω \ B 1 δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ≤ c λ 1 ( n − 2 ) ∕ 2 ∫ R n δ a 2 , λ 2 n + 2 n − 2 ≤ c λ 1 ( n − 2 ) ∕ 2 λ 2 ( n − 2 ) ∕ 2 .

Concerning the other one, using the formula of R a 1 , λ 1 1 , we have to distinguish three cases.

  • For n = 3 , we remark that, in this case, ∣ x − a ∣ δ a , λ ≤ c ∕ λ . Thus, it holds

    ∫ B 1 R a 1 , λ 1 1 ( x ) δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ≤ c λ 1 ∫ R n δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 + ∫ B 1 ∣ x − a 1 ∣ δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ≤ c λ 1 ε 12 + c λ 1 ∫ R n δ a 2 , λ 2 n + 2 n − 2 ≤ c λ 1 ε 12 + c λ 1 λ 2 .

  • For n ≥ 5 , it holds that

    ∫ B 1 R a 1 , λ 1 1 ( x ) δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ≤ c λ 1 2 ∫ R n δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 + c ∫ B 1 ∣ x − a 1 ∣ 2 δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ≤ c ε 12 λ 1 2 + c ∣ a 1 − a 2 ∣ 2 ∫ R n δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 + c ∫ Ω ∣ x − a 2 ∣ 2 δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 .

    Note that

    (A2) ∫ Ω ∣ x − a 2 ∣ 2 δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 = ∫ Ω c λ 2 3 ∕ 2 ∣ x − a 2 ∣ δ a 2 , λ 2 n − 1 n − 2 δ a 1 , λ 1 ≤ c λ 2 3 ∕ 2 ∫ Ω 1 ∣ x − a 2 ∣ 2 n ∕ 3 3 ∕ ( 2 n ) ∫ Ω δ a 2 , λ 2 n − 1 n − 2 δ a 1 , λ 1 2 n 2 n − 3 ( 2 n − 3 ) ∕ ( 2 n ) ≤ c λ 2 3 ∕ 2 ε 12 .

  • For n = 4 ,

    ∫ B 1 R a 1 , λ 1 1 ( x ) δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 ≤ c ln λ 1 λ 1 2 ∫ R n δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 + ∫ B 1 ∣ x − a 1 ∣ 2 ∣ ln ∣ x − a 1 ∣ ∣ δ a 2 , λ 2 n + 2 n − 2 δ a 1 , λ 1 .

    Concerning the second integral, observe that, if ∣ a 1 − a 2 ∣ ≥ e − 4 , it follows that ∣ x − a 2 ∣ ≥ e − 4 ∕ 2 for each x ∈ B 1 , and therefore,

    ∫ B 1 ∣ x − a 1 ∣ 2 ∣ ln ∣ x − a 1 ∣ ∣ δ a 2 , λ 2 3 δ a 1 , λ 1 ≤ c λ 2 3 ∫ B 1 ∣ x − a 1 ∣ 2 ∣ ln ∣ x − a 1 ∣ δ a 1 , λ 1 .

    In the other case, i.e., ∣ a 1 − a 2 ∣ ≤ e − 4 , it follows that ∣ x − a 2 ∣ ≤ ( 3 ∕ 2 ) e − 4 , and therefore, since the function : t ↦ − t 2 ln t is increasing and convex on the set ( 0 , e − 4 ) , we deduce that

    ∣ x − a 1 ∣ 2 ∣ ln ∣ x − a 1 ∣ ∣ ≤ c ∣ x − a 2 ∣ 2 ∣ ln ∣ x − a 2 ∣ ∣ + c ∣ a 2 − a 1 ∣ 2 ∣ ln ∣ a 2 − a 1 ∣ ∣ .

    Thus, the integral becomes

    ∫ B 1 ∣ x − a 1 ∣ 2 ∣ ln ∣ x − a 1 ∣ ∣ δ a 2 , λ 2 3 δ a 1 , λ 1 ≤ c ∫ B 1 ∣ x − a 2 ∣ 2 ∣ ln ∣ x − a 2 ∣ ∣ δ a 2 , λ 2 3 δ a 1 , λ 1 + c ∣ a 2 − a 1 ∣ 2 ∣ ln ∣ a 2 − a 1 ∣ ∣ ∫ B 1 δ a 2 , λ 2 3 δ a 1 , λ 1 ≤ ∫ Ω c ∣ ln ∣ x − a 2 ∣ ∣ λ 2 3 ∕ 2 ∣ x − a 2 ∣ ( δ a 2 , λ 2 3 ∕ 2 δ a 1 , λ 1 ) + c ∣ a 2 − a 1 ∣ 2 ∣ ln ∣ a 2 − a 1 ∣ ∣ ε 12 ≤ c λ 2 3 ∕ 2 ε 12 + c ∣ a 2 − a 1 ∣ 2 ∣ ln ∣ a 2 − a 1 ∣ ∣ ε 12 ,

    where we have used the same computations done in (A2).

This completes the proof of the lemma.□

Lemma A.4

Let n ≥ 4 .

(1) Let d 0 be a fixed small positive constant, a ∈ Ω 0 ≔ { x ∈ Ω : d ( x , ∂ Ω ) ≥ d 0 } , and λ be large. Then, we have

(A3) ‖ π δ a , λ ‖ 2 = S n + O ln σ n λ λ 2 , ⟨ π δ a , λ , φ ⟩ = O ln σ n λ λ m , w i t h m = 2 , i f φ = λ ∂ ( π δ a , λ ) ∕ ∂ λ , m = 3 , i f φ = λ − 1 ∂ ( π δ a , λ ) ∕ ∂ a ,

(A4) λ ∂ π δ a , λ ∂ λ 2 = c + O ln σ n λ λ 2 , λ ∂ π δ a , λ ∂ λ , 1 λ ∂ π δ a , λ ∂ a = O ln σ n λ λ 3 ,

(A5) 1 λ ∂ π δ a , λ ∂ a j 2 = c ″ + O ln σ n λ λ 3 , 1 λ ∂ δ a , λ ∂ a j , 1 λ ∂ δ a , λ ∂ a ℓ = O ln σ n λ λ 3 ∀ ℓ ≠ j .

(2) Let a 1 , a 2 ∈ Ω 0 , and λ 1 , λ 2 be large so that ε 12 , defined in (14), is small. Then, we have

(A6) ⟨ φ 1 , φ 2 ⟩ = O ( ε 12 ) ∀ φ i ∈ π δ a i , λ i , λ i ∂ π δ a i , λ i ∂ λ i , 1 λ i ∂ π δ a i , λ i ∂ a i j , f o r i ∈ { 1 , 2 } a n d j ∈ { 1 , … , n } .

Proof

Note that the first assertion is proved in (35), (39), and (49), respectively. Concerning Claim (A4), using (9) and (10), we obtain

λ ∂ π δ a , λ ∂ λ 2 = ∫ Ω ( − Δ + V ) λ ∂ π δ a , λ ∂ λ λ ∂ π δ a , λ ∂ λ = n + 2 n − 2 ∫ Ω δ a , λ 4 ∕ ( n − 2 ) λ ∂ δ a , λ ∂ λ λ ∂ δ a , λ ∂ λ − λ ∂ θ a , λ ∂ λ = n + 2 n − 2 ∫ R n δ a , λ 4 ∕ ( n − 2 ) λ ∂ δ a , λ ∂ λ 2 + O ∫ R n \ Ω δ a , λ 2 n ∕ ( n − 2 ) + ∫ Ω δ a , λ ( n + 2 ) ∕ ( n − 2 ) θ a , λ .

Thus, (34) and easy computations imply the proof of the first equality in (A4). For the other equality, observe that

⟨ λ ∂ π δ a , λ ∂ λ , 1 λ ∂ π δ a , λ ∂ a ⟩ = ∫ Ω ( − Δ + V ) λ ∂ π δ a , λ ∂ λ 1 λ ∂ π δ a , λ ∂ a = n + 2 n − 2 ∫ Ω δ a , λ 4 n − 2 λ ∂ δ a , λ ∂ λ 1 λ ∂ δ a , λ ∂ a − 1 λ ∂ θ a , λ ∂ a .

Let B ≔ B ( a , d 0 ) , by oddness, it holds that

∫ Ω δ a , λ 4 n − 2 λ ∂ δ a , λ ∂ λ 1 λ ∂ δ a , λ ∂ a = ∫ Ω \ B δ a , λ 4 n − 2 λ ∂ δ a , λ ∂ λ 1 λ ∂ δ a , λ ∂ a = O 1 λ n + 1 .

For the other integral, since a ∈ Ω 0 , we obtain

n + 2 n − 2 ∫ Ω δ a , λ 4 n − 2 λ ∂ δ a , λ ∂ λ 1 λ ∂ θ a , λ ∂ a = ∫ Ω ( − Δ + V ) λ ∂ π δ a , λ ∂ λ 1 λ ∂ θ a , λ ∂ a = ∫ Ω λ ∂ π δ a , λ ∂ λ ( − Δ + V ) 1 λ ∂ θ a , λ ∂ a − ∫ ∂ Ω ∂ ∂ ν λ ∂ π δ a , λ ∂ λ 1 λ ∂ θ a , λ ∂ a = ∫ Ω λ ∂ π δ a , λ ∂ λ V 1 λ ∂ δ a , λ ∂ a + O 1 λ n − 1 .

To complete the proof, using (9)–(11) and the fact that ∣ ∂ δ a , λ ∕ ∂ a ∣ ≤ c δ a , λ ∕ ∣ x − a ∣ , we obtain

∫ Ω \ B λ ∂ π δ a , λ ∂ λ V 1 λ ∂ δ a , λ ∂ a ≤ c ∫ Ω \ B δ a , λ 1 λ ∂ δ a , λ ∂ a ≤ c λ n − 1 , ∫ B λ ∂ π δ a , λ ∂ λ V 1 λ ∂ δ a , λ ∂ a = V ( a ) ∫ B ( a , d 0 ) ∂ δ a , λ ∂ λ ∂ δ a , λ ∂ a + O ∫ B 1 λ δ a , λ 2 + ∫ B R a , λ 1 ( x ) δ a , λ 2 λ ∣ x − a ∣ ≤ c ln σ n λ λ 3 .

This completes the proof of assertion (A4).

The proof of (A5) can be done in the same way than the proof of (A4). Thus, we will omit it here.

Concerning assertion (A6), note that (9) and easy computations imply that

∣ ( − Δ + V ) φ i ∣ ≤ c δ a i , λ i ( n + 2 ) ∕ ( n − 2 ) and ∣ φ i ∣ ≤ c δ a i , λ i , for i ∈ { 1 , 2 } .

Thus, using the last inequality of (33), we deduce that

∣ ⟨ φ 1 , φ 2 ⟩ ∣ = ∫ Ω ( ( − Δ + V ) φ 1 ) φ 2 ≤ c ∫ Ω δ a 1 , λ 1 ( n + 2 ) ∕ ( n − 2 ) δ a 2 , λ 2 ≤ c ε 12 .

This completes the proof of assertion (A6).□

References

[1] M. Bahrami, A. Grobardt, S. Donadi, and A. Bassi, The Schrodinger-Newton equation and its foundations, New J. Phys. 16 (2014), no. 11, 115007, DOI: https://doi.org/10.1088/1367-2630/16/11/115007. 10.1088/1367-2630/16/11/115007Search in Google Scholar

[2] O. Bokanowski, J. L. Lopez, and J. Soler, On an exchange interaction model for quantum transport: the Schrodinger-Poisson-Slater system, Math. Models Methods Appl. Sci. 13 (2003), no. 10, 1397–1412, DOI: https://doi.org/10.1142/S0218202503002969. 10.1142/S0218202503002969Search in Google Scholar

[3] R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativity Gravitation 28 (1996), no. 5, 581–600, DOI: https://doi.org/10.1007/BF02105068. 10.1007/BF02105068Search in Google Scholar

[4] O. Druet and E. Hebey, Elliptic equations of Yamabe type, Int. Math. Res. Surv. 1 (2005), 1–113, DOI: https://doi.org/10.1155/IMRS.2005.1. 10.1155/IMRS.2005.1Search in Google Scholar

[5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477, DOI: https://doi.org/10.1002/cpa.3160360405. 10.1002/cpa.3160360405Search in Google Scholar

[6] A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294, DOI: https://doi.org/10.1002/cpa.3160410302. 10.1002/cpa.3160410302Search in Google Scholar

[7] H. Alharbi, H. Alkhuzayyim, M. Ben Ayed, and K. El Mehdi, Interior peak solutions for a semilinear Dirichlet problem, Axioms 14 (2025), 58, DOI: https://doi.org/10.3390/axioms14010058. 10.3390/axioms14010058Search in Google Scholar

[8] L. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271–297, DOI: https://doi.org/10.1002/cpa.3160420304. 10.1002/cpa.3160420304Search in Google Scholar

[9] A. Pistoia and T. Weith, Sign-changing bubble-tower solutions in a slightly subcritical semi-linear Dirichlet problem, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 2, 325–340, DOI: https://doi.org/10.1016/j.anihpc.2006.03.002. 10.1016/j.anihpc.2006.03.002Search in Google Scholar

[10] J. Wei and S. Yan, Bubble solutions for an elliptic problem with critical growth in exterior domain, Nonlinear Anal. 119 (2015), 46–61, DOI: https://doi.org/10.1016/j.na.2014.08.007. 10.1016/j.na.2014.08.007Search in Google Scholar

[11] A. Pistoia and G. Vaira, Clustering phenomena for linear perturbation of the Yamabe equation. Partial differential equations arising from physics and geometry, in: M. Ben Ayed, M. A. Jendoubi, Y. Rebai, H. Riahi, H. Zaag (Eds), Partial Differential Equations Arising from Physics and Geometry: A Volume in Memory of Abbas Bahri, London Mathematical Society Lecture Note Series, vol. 450, Cambridge University Press, Cambridge, 2019, pp. 311–331. 10.1017/9781108367639.009Search in Google Scholar

[12] F. Robert and J. Vétois, Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non-locally conformally at manifolds, J. Differential Geom. 98 (2014), no. 2, 349–356, DOI: https://doi.org/10.4310/jdg/1406552253. 10.4310/jdg/1406552253Search in Google Scholar

[13] A. Bahri, Critical points at infinity in some variational problems, Pitman Research Notes in Mathematics Series, vol. 182, Longman Scientific & Technical, Harlow; co-published in the United States with John Wiley & Sons, Inc., New York, 1989. Search in Google Scholar

[14] Y. Dammak, A non-existence result for low energy sign-changing solutions of the Brezis-Nirenberg problem in dimensions 4, 5 and 6, J. Differential Equations 263 (2017), no. 11, 7559–7600, DOI: https://doi.org/10.1016/j.jde.2017.08.020. 10.1016/j.jde.2017.08.020Search in Google Scholar

[15] K. El Mehdi and F. Mohamed Salem, Interior bubbling solutions for an elliptic equation with slightly subcritical nonlinearity, Mathematics 11 (2023), no. 6, 1471, DOI: https://doi.org/10.3390/math11061471. 10.3390/math11061471Search in Google Scholar

[16] A. Bahri, Y. Li, and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 67–93, DOI: https://doi.org/10.1007/BF01190892. 10.1007/BF01190892Search in Google Scholar

[17] M. Ben Ayed, K. El Mehdi, and F. Mohamed Salem, Inerior multi-peak solution for a slightly subcritical nonlinear Neumann equation, Symmetry 16 (2024), no. 3, 291, DOI: https://doi.org/10.3390/sym16030291. 10.3390/sym16030291Search in Google Scholar
Received: 2025-01-16
Revised: 2025-04-27
Accepted: 2025-05-06
Published Online: 2025-06-05

© 2025 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. On I-convergence of nets of functions in fuzzy metric spaces
  2. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  3. Forbidden subgraphs of TI-power graphs of finite groups
  4. Finite group with some c#-normal and S-quasinormally embedded subgroups
  5. Classifying cubic symmetric graphs of order 88p and 88p 2
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Simplicial complexes defined on groups
  8. Further results on permanents of Laplacian matrices of trees
  9. Algebra
  10. Classes of modules closed under projective covers
  11. On the dimension of the algebraic sum of subspaces
  12. Green's graphs of a semigroup
  13. On an uncertainty principle for small index subgroups of finite fields
  14. On a generalization of I-regularity
  15. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  16. The hyperbolic CS decomposition of tensors based on the C-product
  17. On weakly classical 1-absorbing prime submodules
  18. Equational characterizations for some subclasses of domains
  19. Algebraic Geometry
  20. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  21. Embedding of lattices and K3-covers of an Enriques surface
  22. Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras
  23. Applications in Computer and Information Sciences
  24. Dynamics of particulate emissions in the presence of autonomous vehicles
  25. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  26. Category Theory
  27. The higher mapping cone axiom
  28. Combinatorics and Graph Theory
  29. 𝕮-inverse of graphs and mixed graphs
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Some new bounds on resolvent energy of a graph
  32. Coloring the vertices of a graph with mutual-visibility property
  33. Ring graph induced by a ring endomorphism
  34. A note on the edge general position number of cactus graphs
  35. Complex Analysis
  36. Some results on value distribution concerning Hayman's alternative
  37. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  38. A new result for entire functions and their shifts with two shared values
  39. On a subclass of multivalent functions defined by generalized multiplier transformation
  40. Singular direction of meromorphic functions with finite logarithmic order
  41. Growth theorems and coefficient bounds for g-starlike mappings of complex order λ
  42. Refinements of inequalities on extremal problems of polynomials
  43. Control Theory and Optimization
  44. Averaging method in optimal control problems for integro-differential equations
  45. On superstability of derivations in Banach algebras
  46. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  47. Observability on the classes of non-nilpotent solvable three-dimensional Lie groups
  48. Differential Equations
  49. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  50. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  51. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  52. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  53. Regularity of weak solutions to the 3D stationary tropical climate model
  54. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  55. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  56. Bubbles clustered inside for almost-critical problems
  57. Existence and multiplicity of positive solutions for multiparameter periodic systems
  58. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  59. On a nonlinear boundary value problems with impulse action
  60. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  61. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  62. Infinitely many solutions for a class of Kirchhoff-type equations
  63. Real and non-real eigenvalues of regular indefinite Sturm–Liouville problems
  64. Existence of global solutions to a semilinear thermoelastic system in three dimensions
  65. Limiting profile of positive solutions to heterogeneous elliptic BVPs with nonlinear flux decaying to negative infinity on a portion of the boundary
  66. Morse index of circular solutions for repulsive central force problems on surfaces
  67. Differential Geometry
  68. On tangent bundles of Walker four-manifolds
  69. Pedal and negative pedal surfaces of framed curves in the Euclidean 3-space
  70. Discrete Mathematics
  71. Eventually monotonic solutions of the generalized Fibonacci equations
  72. Dynamical Systems Ergodic Theory
  73. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  74. A note on weighted measure-theoretic pressure
  75. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  76. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  77. Functional Analysis
  78. Spectrum boundary domination of semiregularities in Banach algebras
  79. Approximate multi-Cauchy mappings on certain groupoids
  80. Investigating the modified UO-iteration process in Banach spaces by a digraph
  81. Tilings, sub-tilings, and spectral sets on p-adic space
  82. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  83. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  84. q-Stirling sequence spaces associated with q-Bell numbers
  85. Chlodowsky variant of Bernstein-type operators on the domain
  86. Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol
  87. Characterization of derivations on strongly double triangle subspace lattice algebras by local actions
  88. Fixed point approaches to the stability of Jensen’s functional equation
  89. Geometry
  90. The regularity of solutions to the Lp Gauss image problem
  91. Solving the quartic by conics
  92. Group Theory
  93. On a question of permutation groups acting on the power set
  94. A characterization of the translational hull of a weakly type B semigroup with E-properties
  95. Harmonic Analysis
  96. Eigenfunctions on an infinite Schrödinger network
  97. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  98. Subharmonic functions and associated measures in ℝn
  99. Mathematical Logic, Model Theory and Foundation
  100. A topology related to implication and upsets on a bounded BCK-algebra
  101. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  102. Number Theory
  103. Fibonacci vector and matrix p-norms
  104. Recurrence for probabilistic extension of Dowling polynomials
  105. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  106. The number of rational points of some classes of algebraic varieties over finite fields
  107. Classification and irreducibility of a class of integer polynomials
  108. Decompositions of the extended Selberg class functions
  109. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  110. Fibonacci Cartan and Lucas Cartan numbers
  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
  113. Numerical Methods
  114. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  115. A kind of univariate improved Shepard-Euler operators
  116. Probability and Statistics
  117. Statistical inference and data analysis of the record-based transmuted Burr X model
  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  120. Real Analysis
  121. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  122. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  123. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  125. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  126. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  127. Topology
  128. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  130. On SI2-convergence in T0-spaces
  131. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
https://doi.org/10.1515/math-2025-0163
Keywords for this article
partial differential equations; variational analysis; nonlinear analysis; critical Sobolev exponent
Creative Commons
BY 4.0

Articles in the same Issue

  1. On I-convergence of nets of functions in fuzzy metric spaces
  2. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  3. Forbidden subgraphs of TI-power graphs of finite groups
  4. Finite group with some c#-normal and S-quasinormally embedded subgroups
  5. Classifying cubic symmetric graphs of order 88p and 88p 2
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Simplicial complexes defined on groups
  8. Further results on permanents of Laplacian matrices of trees
  9. Algebra
  10. Classes of modules closed under projective covers
  11. On the dimension of the algebraic sum of subspaces
  12. Green's graphs of a semigroup
  13. On an uncertainty principle for small index subgroups of finite fields
  14. On a generalization of I-regularity
  15. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  16. The hyperbolic CS decomposition of tensors based on the C-product
  17. On weakly classical 1-absorbing prime submodules
  18. Equational characterizations for some subclasses of domains
  19. Algebraic Geometry
  20. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  21. Embedding of lattices and K3-covers of an Enriques surface
  22. Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras
  23. Applications in Computer and Information Sciences
  24. Dynamics of particulate emissions in the presence of autonomous vehicles
  25. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  26. Category Theory
  27. The higher mapping cone axiom
  28. Combinatorics and Graph Theory
  29. 𝕮-inverse of graphs and mixed graphs
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Some new bounds on resolvent energy of a graph
  32. Coloring the vertices of a graph with mutual-visibility property
  33. Ring graph induced by a ring endomorphism
  34. A note on the edge general position number of cactus graphs
  35. Complex Analysis
  36. Some results on value distribution concerning Hayman's alternative
  37. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  38. A new result for entire functions and their shifts with two shared values
  39. On a subclass of multivalent functions defined by generalized multiplier transformation
  40. Singular direction of meromorphic functions with finite logarithmic order
  41. Growth theorems and coefficient bounds for g-starlike mappings of complex order λ
  42. Refinements of inequalities on extremal problems of polynomials
  43. Control Theory and Optimization
  44. Averaging method in optimal control problems for integro-differential equations
  45. On superstability of derivations in Banach algebras
  46. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  47. Observability on the classes of non-nilpotent solvable three-dimensional Lie groups
  48. Differential Equations
  49. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  50. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  51. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  52. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  53. Regularity of weak solutions to the 3D stationary tropical climate model
  54. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  55. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  56. Bubbles clustered inside for almost-critical problems
  57. Existence and multiplicity of positive solutions for multiparameter periodic systems
  58. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  59. On a nonlinear boundary value problems with impulse action
  60. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  61. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  62. Infinitely many solutions for a class of Kirchhoff-type equations
  63. Real and non-real eigenvalues of regular indefinite Sturm–Liouville problems
  64. Existence of global solutions to a semilinear thermoelastic system in three dimensions
  65. Limiting profile of positive solutions to heterogeneous elliptic BVPs with nonlinear flux decaying to negative infinity on a portion of the boundary
  66. Morse index of circular solutions for repulsive central force problems on surfaces
  67. Differential Geometry
  68. On tangent bundles of Walker four-manifolds
  69. Pedal and negative pedal surfaces of framed curves in the Euclidean 3-space
  70. Discrete Mathematics
  71. Eventually monotonic solutions of the generalized Fibonacci equations
  72. Dynamical Systems Ergodic Theory
  73. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  74. A note on weighted measure-theoretic pressure
  75. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  76. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  77. Functional Analysis
  78. Spectrum boundary domination of semiregularities in Banach algebras
  79. Approximate multi-Cauchy mappings on certain groupoids
  80. Investigating the modified UO-iteration process in Banach spaces by a digraph
  81. Tilings, sub-tilings, and spectral sets on p-adic space
  82. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  83. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  84. q-Stirling sequence spaces associated with q-Bell numbers
  85. Chlodowsky variant of Bernstein-type operators on the domain
  86. Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol
  87. Characterization of derivations on strongly double triangle subspace lattice algebras by local actions
  88. Fixed point approaches to the stability of Jensen’s functional equation
  89. Geometry
  90. The regularity of solutions to the Lp Gauss image problem
  91. Solving the quartic by conics
  92. Group Theory
  93. On a question of permutation groups acting on the power set
  94. A characterization of the translational hull of a weakly type B semigroup with E-properties
  95. Harmonic Analysis
  96. Eigenfunctions on an infinite Schrödinger network
  97. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  98. Subharmonic functions and associated measures in ℝn
  99. Mathematical Logic, Model Theory and Foundation
  100. A topology related to implication and upsets on a bounded BCK-algebra
  101. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  102. Number Theory
  103. Fibonacci vector and matrix p-norms
  104. Recurrence for probabilistic extension of Dowling polynomials
  105. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  106. The number of rational points of some classes of algebraic varieties over finite fields
  107. Classification and irreducibility of a class of integer polynomials
  108. Decompositions of the extended Selberg class functions
  109. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  110. Fibonacci Cartan and Lucas Cartan numbers
  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
  113. Numerical Methods
  114. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  115. A kind of univariate improved Shepard-Euler operators
  116. Probability and Statistics
  117. Statistical inference and data analysis of the record-based transmuted Burr X model
  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  120. Real Analysis
  121. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  122. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  123. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  125. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  126. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  127. Topology
  128. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  130. On SI2-convergence in T0-spaces
  131. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 21.1.2026 from https://www.degruyterbrill.com/document/doi/10.1515/math-2025-0163/html
Scroll to top button