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The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains

  • Jinfeng Xue and Libo Wang EMAIL logo
Published/Copyright: June 20, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we study a p-Laplacian Schrödinger-Poisson system involving a parameter q ≠ 0 in bounded domains. By using the Nehari manifold and the fibering method, we obtain the non-existence and multiplicity of nontrivial solutions. On one hand, there exists q * > 0 such that only the trivial solution is admitted for q ∈ ( q * , + ∞ ) . On the other hand, there are two positive solutions existing for q ∈ ( 0 , q 0 * + ε ) , where ε > 0 and q 0 * + ε < q * . In particular, q * and q 0 * correspond to the supremum for the nonlinear generalized Rayleigh quotients, respectively. The specific form of the nonlinear generalized Rayleigh quotients is calculated. Moreover, it is worth mentioning that we also obtain the qualitative properties associated with the energy level of the solutions.

Keywords: Schrödinger-Poisson system; variational methods; p-Laplacian; fibering method
MSC 2010: 35J20; 35J47; 35J66

1 Introduction

This article studies the following p-Laplacian Schrödinger-Poisson system:

(1.1) − Δ p u + ∣ u ∣ p − 2 u + q 2 ϕ u = ∣ u ∣ γ − 2 u in Ω , − Δ ϕ = 4 π u 2 in Ω , u = 0 , ϕ = 0 on ∂ Ω ,

which can be regarded as an extension of the classical Schrödinger-Poisson system

(1.2) − Δ u + u + q 2 ϕ u = ∣ u ∣ γ − 2 u in Ω , − Δ ϕ = 4 π u 2 in Ω , u = 0 , ϕ = 0 on ∂ Ω ,

where Ω ⊂ R 3 is a bounded domain with smooth boundary, and 2 ≤ p < γ < 3 .

As we all know, systems similar to (1.2) have been investigated extensively, see [1–5]. In particular, the non-existence and multiplicity of nontrivial solutions of system (1.2) have been proved in [2]. From a physical point of view, systems like (1.2) describe a quantum particle’s interaction with an electromagnetic field in quantum mechanics models [6–8] and semiconductor theory [9,10].

Recently, the p-Laplacian Schrödinger-Poisson system has received the attention of some authors, see [11,12]. The existence of nontrivial solutions of the p-Laplacian Schrödinger-Poisson system was obtained by using the mountain pass theorem in [11,12]. Moreover, we find that a type of p-Laplacian Kirchhoff-Schrödinger-Poisson system is studied on the Heisenberg group in [13], and the existence and multiplicity results for the above systems are established by using the concentration-compactness principle in bounded domains. Hence, inspired by [14], we concentrate on the geometric properties of the solutions by fibering map and Nehari manifold, which is helpful to study the non-existence and multiplicity of nontrivial solutions of system (1.1). As a result, we obtain two solutions which are positive by the maximum principle, when q is taken in an appropriate range. Meanwhile, we point out that similar results have been obtained in some nonlinear problems in [15–17] as well.

Now, we give the norm of ϕ and u here and throughout the article: ϕ ∈ H 0 1 ( Ω ) with the norm

‖ ϕ ‖ H = ∫ Ω ∣ ∇ ϕ ∣ 2 d x 1 2

and u ∈ W 0 1 , p ( Ω ) with the norm

‖ u ‖ = ( ‖ ∇ u ‖ p p + ‖ u ‖ p p ) 1 p .

In particular, we denote L p -norm by ‖ u ‖ p . Furthermore, denote the unique solution of the second equation in system (1.1) by ϕ u ∈ H 0 1 ( Ω ) for every u ∈ W 0 1 , p ( Ω ) , which can be obtained by the Lax-Milgram theorem. Since ϕ u satisfies

(1.3) − Δ ϕ u = 4 π u 2 ,

for every u ∈ W 0 1 , p ( Ω ) , from (1.3) we have

(1.4) 4 π ∫ Ω ϕ u u 2 = ‖ ϕ u ‖ H 2 .

In particular, the integral of the whole article is integrated on Ω , and we will not specifically speak later.

We represent the unique solution of the second equation of system (1.1) with ϕ u and bring it into the first equation so that the system reduces to a single equation. Then to solve system (1.1) is equivalent to study the single equation

(1.5) − Δ p u + ∣ u ∣ p − 2 u + q 2 ϕ u u = ∣ u ∣ γ − 2 u in Ω ,

containing the nonlocal term ϕ u . For these reasons, from now on, whenever we refer to the solution of system (1.1), we merely mean the solution u of the above equation since ϕ = ϕ u is explicitly determined.

The energy functional

J q ( u ) = 1 p ‖ u ‖ p + q 2 4 ∫ ϕ u u 2 − 1 γ ‖ u ‖ γ γ , u ∈ W 0 1 , p ( Ω ) ,

which is related to equation (1.5), is well defined and C 1 . Motivated by the above works, we simplify to find critical points of J q .

Before presenting theorems explicitly, for simplicity, the results are given and demonstrated only for q > 0 , because no confusion exists here when q is replaced by ∣ q ∣ . More precisely, the most important result in this article can be stated as follows under the foundation of 2 ≤ p < γ < 3 in bounded domains.

Theorem 1.1

Let 2 ≤ p < γ < 3 , there exist positive constants ε , q 0 * , q * satisfying q 0 * + ε < q * such that:

  1. For q > q * , the functional J q has no critical points except the zero function in W 0 1 , p ( Ω ) .

  2. For q ∈ ( 0 , q 0 * + ε ) , the functional J q has two nontrivial critical points u q , w q ∈ W 0 1 , p ( Ω ) . More specifically,

    1. If q ∈ ( 0 , q 0 * ) , one is a mountain pass critical point w q with

      J q ( w q ) > 0 .

      The other is a global minimum point u q with

      J q ( u q ) < 0 .

    2. If q = q 0 * , one is a mountain pass critical point w q 0 * with

      J q 0 * ( w q 0 * ) > 0 .

      The other is a global minimum point u q 0 * with

      J q 0 * ( u q 0 * ) = 0 .

    3. If q ∈ ( q 0 * , q 0 * + ε ) , one is a mountain pass critical point w q with

      J q ( w q ) > J q ( u q ) .

      The other is a local minimum point u q with

      J q ( u q ) > 0 .

In Theorem 1.1, q * and q 0 * are the extreme values of the Nehari manifold method and correspond to the supremum for the nonlinear generalized Rayleigh quotients, respectively (more details for the notion of extremal parameters, see [18]). As Theorem 1.1 shows, we do not state what happens to the solutions when q between q 0 * + ε and q * . Similar to [19], it seems plausible that there exist two positive solutions for q less than q * and close to it, although we have not proved it yet. Furthermore, it is easy to see that whenever q = q 0 * , J q 0 * is non-negative and we obtain a global minimizer at an energy level of zero. However, extra investigation is required to prove why it does not correspond to the zero function.

Motivated by Ilyasov [18] concerning the so-called extremal value for the application of the Nehari manifold method, this article studies the non-existence and multiplicity of the nontrivial solutions of system (1.1). For this purpose, we introduce the fibering method and Nehari manifolds to find critical points of functional J q . By presenting the possible situations of the fiber map ψ q , u ( t ) , the geometry of the functional J q is directly determined under different values of parameter q . In particular, we calculate the specific form of the nonlinear generalized Rayleigh quotients introduced in [18], which, to the best of our knowledge, has not yet appeared in other articles for this p-Laplacian Schrödinger-Poisson system. Moreover, we apply the Mountain pass theorem and Ekeland’s variational principle, and verify that system (1.1) has two nontrivial solutions when q ∈ ( 0 , q 0 * + ε ) in bounded domains.

This article is organized as follows. In Section 2, we give some preliminaries and technical results. In particular, Theorem 1.1 (1) is proved here. Sections 3–5 are devoted to prove Theorem 1.1 (2) when q is in different values.

2 Preliminaries and technical results

In this section, we give some necessary preliminaries and technical results.

Lemma 2.1

For fixed u ∈ W 0 1 , p ( Ω ) , there exists a unique nonnegative solution ϕ u ∈ H 0 1 ( Ω ) solving − Δ ϕ = 4 π u 2 weakly in Ω . Moreover, ‖ ϕ u ‖ H ≤ C ‖ u ‖ 2 .

Proof

For each u ∈ W 0 1 , p ( Ω ) , we define a linear functional T : H 0 1 ( Ω ) → R by T ( v ) = 4 π ∫ u 2 v . Since the embedding W 0 1 , p ( Ω ) ↪ L λ ( Ω ) , ∀ λ ∈ [ 1 , 6 ) is compact, by using Hölder’s inequality and the embedding H 0 1 ( Ω ) ↪ L 6 ( Ω ) , we deduce that

(2.1) ∣ T ( v ) ∣ ≤ 4 π ∫ ∣ u ∣ 12 5 5 6 ∫ ∣ v ∣ 6 1 6 ≤ C ‖ u ‖ 2 ‖ v ‖ H ,

from which the boundedness of T on H 0 1 ( Ω ) follows. According to the definition of weak solution of (1.3) and the Lax-Milgram theorem, there exists a unique ϕ u ∈ H 0 1 ( Ω ) such that

(2.2) ∫ ∇ ϕ u ∇ v = 4 π ∫ u 2 v , ∀ v ∈ H 0 1 ( Ω ) .

Therefore, combining (2.1) with (2.2) and choosing v = ϕ u , we obtain ‖ ϕ u ‖ H ≤ C ‖ u ‖ 2 .□

Proposition 2.2

For all u ∈ W 0 1 , p ( Ω ) ,

∫ ∣ u ∣ 3 ≤ 1 4 π ‖ ϕ u ‖ H ‖ ∇ u ‖ 2 .

Proof

For each fixed u ∈ W 0 1 , p ( Ω ) , multiplying equation (1.3) by ∣ u ∣ ∈ W 0 1 , p ( Ω ) and integrating on Ω , using Hölder’s inequality and the embedding W 0 1 , p ( Ω ) ↪ H 0 1 ( Ω ) , we obtain

4 π ∫ ∣ u ∣ 3 = ∫ ∇ ϕ u ∇ ∣ u ∣ ≤ ‖ ∇ ϕ u ‖ 2 ‖ ∇ u ‖ 2 .

It follows that

□ ∫ ∣ u ∣ 3 ≤ 1 4 π ‖ ϕ u ‖ H ‖ ∇ u ‖ 2 .

Proposition 2.3

There exists ρ > 0 and M > 0 satisfying

J q ( u ) ≥ M for a l l q > 0 and u ∈ W 0 1 , p ( Ω ) w i t h ‖ u ‖ = ρ .

Proof

The conclusion easily follows from

(2.3) J q ( u ) ≥ 1 p ‖ u ‖ p − 1 γ ‖ u ‖ γ γ ≥ 1 p ‖ u ‖ p − C ‖ u ‖ γ ,

where C > 0 is the Sobolev embedding constant.□

Now, we study the geometry of the functional J q . Observing ϕ t u = t 2 ϕ u and defining ψ q , u ( t ) = J q ( t u ) with ψ q , u : [ 0 , ∞ ) → R , we have

ψ q , u ( t ) = t p p ‖ u ‖ p + q 2 t 4 4 ∫ ϕ u u 2 − t γ γ ‖ u ‖ γ γ .

For brevity, we will also use the notation ψ ≔ ψ q , u whenever q and u are fixed. A simple analysis is as follows.

Proposition 2.4

For each q > 0 and u ∈ W 0 1 , p ( Ω ) \ { 0 } , the graph of ψ has three possibilities on ( 0 , + ∞ ) :

  1. The function ψ has two critical points, i.e., 0 < t q − ( u ) < t q + ( u ) . Furthermore, t q − ( u ) is a local maximum point with ψ ″ ( t q − ( u ) ) < 0 and t q + ( u ) is a local minimum point with ψ ″ ( t q + ( u ) ) > 0 .

  2. The function ψ has one critical point, i.e., 0 < t q ( u ) . Furthermore, ψ ″ ( t q ( u ) ) = 0 and ψ is increasing.

  3. The function ψ has no critical points and is increasing.

Note that (i) occurs for q small and (iii) for q large (Figure 1).

Figure 1 Fiber map graphs for Proposition 2.4.
Figure 1

Fiber map graphs for Proposition 2.4.

Now, we introduce the Nehari manifolds associated with the functional J q , namely,

N q = { u ∈ W 0 1 , p ( Ω ) \ { 0 } : ψ ′ ( 1 ) = 0 } .

For u ∈ N q ,

(2.4) ‖ u ‖ p ≤ ‖ u ‖ p + q 2 ∫ ϕ u u 2 ≤ C ‖ u ‖ γ .

Then, we know that

(2.5) ∃ C ˜ > 0 such that for all q > 0 , u ∈ N q one has ‖ u ‖ ≥ C ˜ ,

that is, all the Nehari manifolds are uniformly bounded away from zero in q .

Moreover, the Nehari set can be divided into three disjoint sets:

N q = N q + ∪ N q 0 ∪ N q − ,

where

N q + = { u ∈ N q : ψ ″ ( 1 ) > 0 } ,

N q 0 = { u ∈ N q : ψ ″ ( 1 ) = 0 } ,

N q − = { u ∈ N q : ψ ″ ( 1 ) < 0 } .

As a result of the implicit function theorem, one obtains:

Proposition 2.5

Consider the disjoint sets:

  1. N q + , N q − are C 1 manifolds of codimension 1 in W 0 1 , p ( Ω ) if N q + , N q − are non-empty;

  2. u ∈ N q + ∪ N q − is a critical point for the functional J q , equivalently, u is a critical point of the constrained functional ( J q ) ∣ N q + ∪ N q − : N q + ∪ N q − → R .

From the previous discussion, we have t u ∈ N q 0 for a fixed u ∈ W 0 1 , p ( Ω ) if and only if ψ q , t u ′ ( 1 ) = ψ q , t u ″ ( 1 ) = 0 . Indeed, this is equivalent to the system

(2.6) t p − 1 ‖ u ‖ p + q 2 t 3 ∫ ϕ u u 2 − t γ − 1 ‖ u ‖ γ γ = 0 , ( p − 1 ) t p − 2 ‖ u ‖ p + 3 q 2 t 2 ∫ ϕ u u 2 − ( γ − 1 ) t γ − 2 ‖ u ‖ γ γ = 0 .

After direct calculation, we obtain the unique solution with respect to the variables q and t , which are

t ( u ) = ( 4 − p ) ‖ u ‖ p ( 4 − γ ) ‖ u ‖ γ γ 1 γ − p

and

(2.7) q ( u ) = C p ‖ u ‖ γ γ ( 4 − p ) 2 ( γ − p ) ‖ u ‖ p ( 4 − γ ) 2 ( γ − p ) ‖ ϕ u ‖ H ,

where

(2.8) C p = 4 − p 4 − γ γ − 4 γ − p − 4 − p 4 − γ p − 4 γ − p 4 π 1 2 .

In particular, there exists a connection between q and t ,

t ( u ) = q 2 ( u ) C ‖ ϕ u ‖ H 2 ‖ u ‖ γ γ 1 γ − 4 ,

where C is a positive constant. Now, we define the extremal value

q * = sup { q ( u ) : u ∈ W 0 1 , p ( Ω ) \ { 0 } } .

Lemma 2.6

The function W 0 1 , p ( Ω ) \ { 0 } ∋ u ↦ q ( u ) defined in (2.7) is 0-homogeneous. Moreover, q * < ∞ .

Proof

It is obvious that q ( u ) is 0-homogeneous. Next, we prove q * < ∞ . In fact, from the interpolation inequality for γ ∈ ( 2 , 3 ) , we obtain

(2.9) ‖ u ‖ γ γ ≤ ‖ u ‖ 2 6 − 2 γ ‖ u ‖ 3 3 γ − 6

for all u ∈ W 0 1 , p ( Ω ) . From Proposition 2.2 and W 0 1 , p ( Ω ) ↪ H 0 1 ( Ω ) , L p ( Ω ) ↪ L 2 ( Ω ) , we have

(2.10) ‖ u ‖ γ γ ( 4 − p ) ≤ ‖ u ‖ 2 ( 6 − 2 γ ) ( 4 − p ) ‖ u ‖ 3 ( 3 γ − 6 ) ( 4 − p ) ≤ C ‖ u ‖ 2 ( 6 − 2 γ ) ( 4 − p ) ‖ ∇ u ‖ 2 ( γ − 2 ) ( 4 − p ) ‖ ϕ u ‖ H ( γ − 2 ) ( 4 − p ) ≤ C ‖ u ‖ p ( 6 − 2 γ ) ( 4 − p ) ‖ ∇ u ‖ p ( γ − 2 ) ( 4 − p ) ‖ ϕ u ‖ H ( γ − 2 ) ( 4 − p ) ≤ C ‖ u ‖ ( 4 − γ ) ( 4 − p ) ‖ ϕ u ‖ H ( γ − 2 ) ( 4 − p ) ,

for some constant C > 0 . Consequently, according to Lemma 2.1 and the definition of q ( u ) , we conclude that

□ q ( u ) ≤ C ‖ u ‖ ( 4 − γ ) ( 4 − p ) 2 ( γ − p ) ‖ ϕ u ‖ H ( γ − 2 ) ( 4 − p ) 2 ( γ − p ) ‖ u ‖ ( 4 − γ ) p 2 ( γ − p ) ‖ ϕ u ‖ H ≤ C ‖ u ‖ ( 4 − γ ) ( 2 − p ) γ − p ‖ ϕ u ‖ H ( γ − 4 ) ( 2 − p ) 2 ( γ − p ) ≤ C ‖ u ‖ ( 4 − γ ) ( 2 − p ) γ − p ‖ u ‖ ( γ − 4 ) ( 2 − p ) γ − p = C .

On the contrary, we have another extremal value which is crucial in the following proof. Under this circumstance, the functional J q is always non-negative. Let us begin by fixing u ∈ W 0 1 , p ( Ω ) \ { 0 } and taking the following system into consideration:

(2.11) ψ q 0 , u ( t 0 ) = t 0 p p ‖ u ‖ p + q 0 2 t 0 4 4 ∫ ϕ u u 2 − t 0 γ γ ‖ u ‖ γ γ = 0 , ψ q 0 , u ′ ( t 0 ) = t 0 p − 1 ‖ u ‖ p + q 0 2 t 0 3 ∫ ϕ u u 2 − t 0 γ − 1 ‖ u ‖ γ γ = 0 .

With regard to the variables t 0 and q 0 , we give the unique solution

t 0 ( u ) = γ ( 4 − p ) ‖ u ‖ p p ( 4 − γ ) ‖ u ‖ γ γ 1 γ − p

and

(2.12) q 0 ( u ) = C 0 , p ‖ u ‖ γ γ ( 4 − p ) 2 ( γ − p ) ‖ u ‖ p ( 4 − γ ) 2 ( γ − p ) ‖ ϕ u ‖ H ,

where

(2.13) C 0 , p = γ ( 4 − p ) p ( 4 − γ ) γ − 4 γ − p − γ ( 4 − p ) p ( 4 − γ ) p − 4 γ − p 4 π 1 2 .

In particular, there exists a connection between q 0 and t 0 ,

t 0 ( u ) = q 0 2 ( u ) C ‖ ϕ u ‖ H 2 ‖ u ‖ γ γ 1 γ − 4 ,

where C is a positive constant. Observe that C 0 , p < C p , where C p is the one shown in (2.8). After that, q 0 ( u ) < q ( u ) . The extremal value was described as

q 0 * = sup { q 0 ( u ) : u ∈ W 0 1 , p ( Ω ) \ { 0 } } .

Remark 1

Since q 0 ( u ) is a multiple of q ( u ) , Lemma 2.6 also holds true for the function q 0 ( u ) .

Proposition 2.7

There exists a positive constant m such that

J q ( u ) ≥ m for a l l q > 0 , u ∈ N q 0 .

Proof

Using equation (2.6) and taking t = 1 , we can deduce that

(2.14) ‖ u ‖ p + q 2 ∫ ϕ u u 2 − ‖ u ‖ γ γ = 0 , ( p − 1 ) ‖ u ‖ p + 3 q 2 ∫ ϕ u u 2 − ( γ − 1 ) ‖ u ‖ γ γ = 0 .

It follows that

q 2 ∫ ϕ u u 2 = ‖ u ‖ γ γ − ‖ u ‖ p and ‖ u ‖ γ γ = p − 4 γ − 4 ‖ u ‖ p .

Then, we have J q ( u ) = ( 4 − p ) ( γ − p ) 4 p γ ‖ u ‖ p for each u ∈ N q 0 . Hence, we complete the proof by (2.5).□

Lemma 2.8

Consider the operator Φ : W 0 1 , p ( Ω ) → H 0 1 ( Ω ) , Φ ( u ) = ϕ u , that is, the solution of the problem − Δ ϕ u = 4 π u 2 in H 0 1 ( Ω ) . Let u n be a sequence satisfying u n ⇀ u in W 0 1 , p ( Ω ) . Then, ϕ u n → ϕ u in H 0 1 ( Ω ) and, as a consequence,

∫ ϕ u n u n 2 → ∫ ϕ u u 2 .

Proof

Define the linear operators T n , T : H 0 1 ( Ω ) → R by

T n ( v ) = 4 π ∫ u n 2 v , T ( v ) = 4 π ∫ u 2 v .

Recall that the embedding W 0 1 , p ( Ω ) into L λ ( Ω ) is compact for 1 ≤ λ < 6 . Then

∫ ∣ u n 2 − u 2 ∣ 6 5 = ∫ ∣ u n + u ∣ 6 5 ∣ u n − u ∣ 6 5 ≤ ∫ ∣ u n + u ∣ 12 5 1 2 ∫ ∣ u n − u ∣ 12 5 1 2 ,

hence u n 2 → u 2 in L 6 5 . Note that

∣ T n ( v ) − T ( v ) ∣ ≤ 4 π ∫ ∣ u n 2 − u 2 ∣ 6 5 5 6 ∫ ∣ v ∣ 6 1 6 ≤ 4 π ‖ u n 2 − u 2 ‖ 6 5 ‖ v ‖ H ,

which means T n converges strongly to T .

From (1.3), we obtain that

(2.15) ∫ ∇ ϕ u ∇ v = 4 π ∫ u 2 v , ∀ v ∈ H 0 1 ( Ω ) ,

(2.16) ∫ ∇ ϕ u n ∇ v = 4 π ∫ u n 2 v , ∀ v ∈ H 0 1 ( Ω ) .

Subtracting (2.15) from (2.16) and choosing v = ϕ u n − ϕ u , we obtain

‖ ϕ u n − ϕ u ‖ H 2 = 4 π ∫ ( u n 2 − u 2 ) ( ϕ u n − ϕ u ) ≤ 4 π ∫ ∣ u n 2 − u 2 ∣ 6 5 5 6 ∫ ∣ ( ϕ u n − ϕ u ) ∣ 6 1 6 ≤ 4 π ‖ u n 2 − u 2 ‖ 6 5 ‖ ϕ u n − ϕ u ‖ H .

Hence, ϕ u n → ϕ u in H 0 1 ( Ω ) .

Conversely, observing that ϕ u n → ϕ u in L 6 ( Ω ) and u n 2 → u 2 in L 6 5 , we have

(2.17) ∫ ϕ u n u n 2 − ∫ ϕ u u 2 = ∫ ϕ u n u n 2 − ∫ ϕ u n u 2 + ∫ ϕ u n u 2 − ∫ ϕ u u 2 ≤ ∫ ϕ u n u n 2 − ∫ ϕ u n u 2 + ∫ ϕ u n u 2 − ∫ ϕ u u 2 ≤ ∫ ∣ u n 2 − u 2 ∣ 6 5 5 6 ∫ ∣ ϕ u n ∣ 6 1 6 + ∫ ∣ u 2 ∣ 6 5 5 6 ∫ ∣ ϕ u n − ϕ u ∣ 6 1 6 ≤ C ‖ u n 2 − u 2 ‖ 6 5 ‖ ϕ u n ‖ H + C ‖ u ‖ 12 5 2 ‖ ϕ u n − ϕ u ‖ H → 0 ,

then we conclude ∫ ϕ u n u n 2 → ∫ ϕ u u 2 . □

Lemma 2.9

The functional J q ( u ) satisfies the following properties:

  1. J q ( u ) is weakly lower semi-continuous and coercive.

  2. J q ( u ) satisfies the Palais-Smale condition.

Proof

To prove (1), we take u n ⇀ u and then by using the compactness of the Sobolev embedding, Brezis-Lieb’s lemma, and Lemma 2.8, we obtain

∫ ∣ u n ∣ γ → ∫ ∣ u ∣ γ , ∫ ϕ u n u n 2 → ∫ ϕ u u 2 .

It follows that J q ( u ) is w.l.s.c.

Conversely, we begin to prove that J q ( u ) is coercive. By using Hölder’s inequality, Young’s inequality, and Proposition 2.2, we obtain

(2.18) ‖ u ‖ 3 3 ≤ 1 4 π ∣ meas ( Ω ) ∣ 1 τ ‖ ∇ ϕ u ‖ 2 ‖ ∇ u ‖ p ≤ 1 4 π 1 τ ∣ meas ( Ω ) ∣ + 1 p ε p ‖ ∇ u ‖ p p + 1 2 ε 2 ‖ ∇ ϕ u ‖ 2 2 = 1 4 π 1 τ ∣ meas ( Ω ) ∣ + 1 p ε p ‖ ∇ u ‖ p p + 1 2 ε 2 ‖ ϕ u ‖ H 2 ,

where

1 τ + 1 p + 1 2 = 1 ,

τ = 2 p p − 2 ∈ ( 6 , + ∞ ) because of p ∈ ( 2 , 3 ) .

We take D q = q 2 16 π − ε p + 2 4 > 0 such that

(2.19) J q ( u ) = 1 2 p ‖ ∇ u ‖ p p + 1 2 p ‖ ∇ u ‖ p p + 1 p ‖ u ‖ p p + q 2 4 ∫ ϕ u u 2 − 1 γ ‖ u ‖ γ γ ≥ 1 2 p ‖ ∇ u ‖ p p + q 2 16 π − ε p + 2 4 ‖ ϕ u ‖ H 2 + 1 p ‖ u ‖ p p + 2 π ε p ‖ u ‖ 3 3 − 1 γ ‖ u ‖ γ γ − ε p ∣ Ω ∣ 2 τ = 1 2 p ‖ u ‖ p − ε p ∣ Ω ∣ 2 τ + D q ‖ ϕ u ‖ H 2 + ∫ f ( ∣ u ∣ ) ,

where

f ( t ) = 1 2 p t p + 2 π ε p t 3 − 1 γ t γ for all t > 0 .

A simple analysis shows that I = inf t > 0 f ( t ) > − ∞ and if f ( t ) < 0 for some t > 0 , then f − 1 ( ( − ∞ , 0 ) ) = ( α , β ) , where 0 < α < β < ∞ .

If I ≥ 0 , since D q > 0 we conclude from (2.19) that

(2.20) J q ( u ) ≥ 1 2 p ‖ u ‖ p − ε p ∣ Ω ∣ 2 τ + D q ‖ ϕ u ‖ H 2 ≥ 1 2 p ‖ u ‖ p − C .

If I < 0 , then

(2.21) J q ( u ) ≥ 1 2 p ‖ u ‖ p − ε p ∣ Ω ∣ 2 τ + D q ‖ ϕ u ‖ H 2 + I meas ( A ) ≥ 1 2 p ‖ u ‖ p − C ,

where A = { x ∈ Ω : u ( x ) ∈ ( α , β ) } . Hence, J q ( u ) is coercive.

Now, we begin to prove that u n admits a converging subsequence. By the boundedness and coercivity of J q ( u ) , it is clear that u n is also bounded. Thanks to the boundedness of { u n } , there exists u 0 ∈ W 0 1 , p ( Ω ) such that, up to subsequences, u n ⇀ u 0 in W 0 1 , p ( Ω ) , then u n → u 0 in L λ ( Ω ) , 1 ≤ λ < 6 because of the previous compact embedding. We evaluate

⟨ J q ′ ( u n ) , u n − u ⟩ = ∫ ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n − u ) + ∫ ∣ u n ∣ p − 2 u n ( u n − u ) + q 2 ∫ ϕ u n u n ( u n − u ) − ∫ ∣ u n ∣ γ − 2 u n ( u n − u ) .

By using Hölder’s inequality, we have:

∫ ϕ u n u n ( u n − u ) ≤ ‖ ϕ u n ‖ 3 ‖ u n ‖ 3 ‖ u n − u ‖ 3 .

By Lemma 2.1, the above expression tends to zero as n → ∞ . Moreover,

∫ ∣ u n ∣ p − 2 u n ( u n − u ) → 0 , ∫ ∣ u n ∣ γ − 2 u n ( u n − u ) → 0 .

Then

∫ ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n − u ) → 0 .

Finally, we conclude u n → u in W 0 1 , p ( Ω ) .□

Remark 2

Lemma 2.9 can be generalized as follows: depending on the smooth dependence of J q ′ on q , we obtain the conclusion that if q n → q and { u n } ⊂ W 0 1 , p ( Ω ) is a sequence such that J q ′ ( u n ) → 0 as n → + ∞ , then { u n } is convergent, up to subsequences.

Finally, we give the following geometrical interpretations of q ( u ) and q 0 ( u ) , which can be easily derived by Proposition 2.4.

Proposition 2.10

For each u ∈ W 0 1 , p ( Ω ) \ { 0 } , q > 0 , there holds

  1. The fiber map ψ has a critical point with second derivative zero at t ( u ) . Furthermore, if 0 < q < q ( u ) , then ψ corresponds to Proposition 2.4 (i) while if q > q ( u ) , ψ corresponds to Proposition 2.4 (iii).

  2. The fiber map ψ has a critical point with zero energy at t 0 ( u ) . Furthermore, if 0 < q < q 0 ( u ) , inf t > 0 ψ ( t ) < 0 while if q > q 0 ( u ) , inf t > 0 ψ ( t ) = 0 .

In addition, the parameter q 0 * possesses such a geometric interpretation that when 0 < q < q 0 * , J q ( u ) < 0 for at least one u ∈ W 0 1 , p ( Ω ) \ { 0 } while q ≥ q 0 * , J q ( u ) ≥ 0 for all u ∈ W 0 1 , p ( Ω ) . It is a consequence that small values q are necessary to prove the existence of a function with a negative value, such that J q has a mountain pass geometry.

Corollary 2.11

If q ≥ q 0 * , then J q ≥ 0 for all u ∈ W 0 1 , p ( Ω ) . Moreover, if q < q 0 * , then there exists u ∈ W 0 1 , p ( Ω ) such that J q < 0 .

Proof

In fact, assume that q ≥ q 0 * . Deriving from Proposition 2.10 (ii), we obtain inf t > 0 ψ ( t ) = 0 . This means that when q > q 0 ( u ) for each u ∈ W 0 1 , p ( Ω ) \ { 0 } , we have J q ( u ) ≥ 0 . Conversely, suppose that q < q 0 * . According to the definition of q 0 * , there exists w ∈ W 0 1 , p ( Ω ) \ { 0 } such that q < q 0 ( w ) < q 0 * . Consequently, we deduce from Proposition 2.10 (ii) that inf t > 0 ψ ( t ) < 0 and then there exists t > 0 such that if u ≔ t w , J q < 0 . □

Corollary 2.12

If q > q * , the functional J q has no critical points except the zero function. Moreover, if q < q * , then N q − ≠ ∅ and N q + ≠ ∅ .

Proof

It suffices to prove that the function ψ has no critical points when q > q * for each u ∈ W 0 1 , p ( Ω ) \ { 0 } . Actually, due to the inequality q ( u ) ≤ q * < q and Proposition 2.10 (i), we can obtain the conclusion easily. Now, assume that q < q * . According to the definition of q * , there exists u ∈ W 0 1 , p ( Ω ) \ { 0 } such that q < q ( u ) < q * . As a result, we deduce from Proposition 2.10 (i) that N q − ≠ ∅ and N q + ≠ ∅ . □

In particular, Theorem 1.1 (1) has been proved here.

3 Existence of solutions for q ∈ ( 0 , q 0 * )

In this section, we will prove Theorem 1.1 (2) (i).

(i) The global minimum solution

Proposition 3.1

For each q ∈ ( 0 , q 0 * ) , − ∞ < inf u ∈ W 0 1 , p ( Ω ) J q ( u ) < 0 .

Proof

From Corollary 2.11, we know that

inf u ∈ W 0 1 , p ( Ω ) J q ( u ) < 0 .

Now, we prove − ∞ < inf u ∈ W 0 1 , p ( Ω ) J q ( u ) . By contradiction, we assume that there exists a sequence { u n } such that J q ( u n ) → − ∞ as n → + ∞ , then ‖ u n ‖ → + ∞ because of (2.3). Without loss of generality, we can infer from (2.21) that

1 2 p ‖ u n ‖ p ≤ J q ( u n ) + ε p ∣ Ω ∣ 2 τ − I meas ( A n ) ,

where A n ⊂ Ω . There is no doubt that it is a contradiction here.□

Proposition 3.2

For q ∈ ( 0 , q 0 * ) , there exists a global minimum u q such that J q ( u q ) < 0 .

Proof

The conclusion can be easily proved by using Proposition 3.1 and Ekeland’s variational principle.□

(ii) The mountain pass solution

Define

c q ≔ inf γ ∈ Γ q max t ∈ [ 0 , 1 ] J q ( γ ( t ) ) > 0 ,

where

Γ q = { γ ∈ C ( [ 0 , 1 ] , W 0 1 , p ( Ω ) ) : γ ( 0 ) = 0 , J q ( γ ( 1 ) ) < 0 } .

Thanks to Corollary 2.11, Γ q is non-empty. The following proposition is deduced from Proposition 2.3 and Lemma 2.9.

Proposition 3.3

For each q ∈ ( 0 , q 0 * ) , there exists w q ∈ W 0 1 , p ( Ω ) \ { 0 } such that J q ( w q ) = c q and J q ′ ( w q ) = 0 . Particularly, J q ( w q ) > J q ( u q ) ∈ ( − ∞ , 0 ) .

4 Existence of solutions for q = q 0 *

In this section, we will prove Theorem 1.1 (2) (ii).

(i) The global minimum solution

Corollary 4.1

There exists a global minimum point u q 0 * ≠ 0 of J q 0 * such that J q 0 * ( u q 0 * ) = 0 .

Proof

Assume that q n ↑ q 0 * as n → ∞ . For each n , there exists u n ≔ u q n , which is a global minimum for J q n and J q n ( u n ) < 0 from Proposition 3.2. Then, we have J q n ′ ( u n ) = 0 for each n . Conversely, observing all the Nehari manifolds are bounded away from zero uniformly in q , we have ‖ u n ‖ ≥ C ˜ for each n . Moreover, we conclude that u n → u ≠ 0 from Remark 2. Since J q n ( u n ) < 0 for each n , passing to the limit as n → ∞ , we obtain J q 0 * ( u ) ≤ 0 . As a consequence, we obtain J q 0 * ( u ) = 0 from Corollary 2.11. Finally, we set u q 0 * ≔ u and the conclusion follows.□

Remark 3

q 0 ( u q 0 * ) = q 0 * follows from the definition of q 0 * and Corollary 4.1. Moreover, q 0 * < q ( u q 0 * ) .

(ii) The mountain pass solution

Define

c q 0 * ≔ inf γ ∈ Γ q 0 * max t ∈ [ 0 , 1 ] J q 0 * ( γ ( t ) ) > 0 ,

where

Γ q 0 * = { γ ∈ C ( [ 0 , 1 ] , W 0 1 , p ( Ω ) ) : γ ( 0 ) = 0 , J q 0 * ( γ ( 1 ) ) = 0 } .

The following proposition is deduced from Proposition 2.3 and Lemma 2.9.

Proposition 4.2

For q = q 0 * , there exists w q 0 * ∈ W 0 1 , p ( Ω ) \ { 0 } such that J q 0 * ( w q 0 * ) = c q 0 * and J q 0 * ′ ( w q 0 * ) = 0 . Particularly, J q 0 * ( w q 0 * ) > J q 0 * ( u q 0 * ) = 0 .

5 Existence of solutions for q ∈ ( q 0 * , q 0 * + ε )

In this section, we will prove Theorem 1.1 (2) (iii).

Consider the following family of constrained minimization problems: For q > 0 ,

J ^ q ≔ inf { J q ( u ) : u ∈ N q + ∪ N q 0 } .

Observe that

J ^ q = inf u ∈ W 0 1 , p ( Ω ) J q ( u ) for all q ∈ ( 0 , q 0 * ] ,

and from Corollary 2.11, we have J ^ q ≥ 0 for q ≥ q 0 * .

(i) The local minimum solution

Proposition 5.1

Given δ > 0 , there exists ε > 0 such that J ^ q < δ for each q ∈ ( q 0 * , q 0 * + ε ) .

Proof

We obtain u q 0 * ∈ N q + as in Corollary 4.1. According to Remark 3, there exists ε 1 > 0 such that q 0 * + ε 1 < q ( u q 0 * ) . If q ↓ q 0 * , then J q ( u q 0 * ) → J q 0 * ( u q 0 * ) = 0 . Conversely, for q ∈ ( q 0 * , q 0 * + ε ) , there exists t q + ( u q 0 * ) such that t q + ( u q 0 * ) u q 0 * ∈ N q + by Proposition 2.4 and Proposition 2.10 (ii). When t q + ( u q 0 * ) → 1 as q ↓ q 0 * , we have

0 ≤ J q ( t q + ( u q 0 * ) u q 0 * ) ≤ J q ( u q 0 * ) → J q 0 * ( u q 0 * ) = 0 , q ↓ q 0 * .

The conclusion follows if we choose ε 2 in such a way that J q ( t q + ( u q 0 * ) u q 0 * ) < δ for each q ∈ ( q 0 * , q 0 * + ε 2 ) . Then, we take ε = min { ε 1 , ε 2 } and complete the proof.□

Recall that by Proposition 2.3, there are positive constants ρ , M satisfying J q ( u ) ≥ M for each ‖ u ‖ = ρ . Without loss of generality, we may assume that ρ < C ˜ , where C ˜ is in such a way that

‖ u ‖ ≥ C ˜ for all q > 0 and u ∈ N q

(see (2.5)).

We choose δ > 0 in Proposition 5.1 in such a way that

(5.1) 0 < δ < min { M , m } ,

where m is the positive constant such that, by Proposition 2.7,

J q ( u ) ≥ m for all q > 0 and u ∈ N q 0 .

Let ε > 0 be as in Proposition 5.1 in correspondence of the above fixed δ > 0 .

Proposition 5.2

For all q ∈ ( q 0 * , q 0 * + ε ) ,

inf { J q ( u ) : ‖ u ‖ ≥ ρ } = J ^ q .

Proof

In fact, we claim that the inequality inf { J q ( u ) : ‖ u ‖ ≥ ρ } ≤ J ^ q holds for ρ < C ˜ . According to Proposition 2.4, we distinguish the following three cases. When the fiber map ψ satisfies Proposition 2.4 (i), we have inf t > ρ ψ ( t ) ≥ J ^ q . When the fiber map ψ satisfies Proposition 2.4 (ii) or (iii), we have inf t > ρ ψ ( t ) = M . As the condition M > δ > J ^ q is satisfied, we finish this proof.□

Corollary 5.3

For q ∈ ( q 0 * , q 0 * + ε ) , there exists u q ∈ N q + such that J q ( u q ) = J ^ q . Particularly, J q ( u q ) > 0 and ‖ u q ‖ ≥ C ˜ > ρ .

Proof

For fixed q ∈ ( q 0 * , q 0 * + ε ) , we take a minimizing sequence { u n } ∈ N q + ∪ N q 0 for J ^ q < δ by Proposition 5.1. From Proposition 2.7 that J q ( u ) ≥ m on N q 0 and m > δ in (5.1), we assume that { u n } ⊂ N q + . Therefore, by Ekeland’s variational principle, we also have J q ′ ( u n ) → 0 . Hence, we deduce from Lemma 2.9 that u n → u in W 0 1 , p ( Ω ) with ‖ u ‖ ≥ C ˜ > ρ . Setting u q ≔ u clearly we obtain that u q ∈ N q + and J q ( u q ) = J ^ q . On account of the fact that q > q 0 * and the definition of q 0 * , we deduce that J q ( u q ) > 0 . □

(ii) The mountain pass solution

Now, for q ∈ ( q 0 * , q 0 * + ε ) , we select ε > 0 such that (5.1) holds for fixed δ > 0 . Owing to J q ( u q ) = J ^ q for u q ∈ N q + (by Corollary 5.3), we define

d q ≔ inf γ ∈ Γ q max t ∈ [ 0 , 1 ] J ( γ ( t ) ) ,

where

Γ q = { γ ∈ C ( [ 0 , 1 ] , W 0 1 , p ( Ω ) ) : γ ( 0 ) = 0 , γ ( 1 ) = u q } .

Proposition 5.4

For q ∈ ( q 0 * , q 0 * + ε ) , there exists w q ∈ W 0 1 , p ( Ω ) \ { 0 } such that J q ( w q ) = d q and J q ′ ( w q ) = 0 . Particularly, J q ( w q ) > J q ( u q ) .

Proof

From Proposition 2.3 and (5.1), for all q > 0 , there exists ρ > 0 and M > 0 satisfying J q ( u ) ≥ M > δ for every u ∈ S ρ = { u ∈ W 0 1 , p ( Ω ) : ‖ u ‖ = ρ } . Conversely, we know that 0 = J q ( 0 ) < δ and J q ( u q ) = J ^ q ≤ δ . Then, we obtain a mountain pass geometry for the functional J q , which can be found in [20]. The proof follows from Lemma 2.9.□

Now, we conclude the proof of Theorem 1.1 (2). The existence of the minimum critical point u q is based on Proposition 3.2, Corollary 4.1, and Corollary 5.3. The existence of a mountain pass critical point w q which satisfies J q ( w q ) > max { 0 , J q ( u q ) } is based on Propositions 3.3, 4.2, and 5.4. Actually, u q and w q are critical points of J q followed by Proposition 2.5, which correspond to the solutions of system (1.1).

Acknowledgments

The authors are grateful to the anonymous referee whose careful reading of this manuscript and valuable comments enhanced the presentation of this manuscript.

  1. Funding information: This research was supported by the Natural Science Foundation of Jilin Province (Grant No. 20210101156JC).

  2. Author contributions: Manuscript preparation: JX; supervision: LW. All authors read and approved the final manuscript.

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

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Received: 2023-10-12
Revised: 2024-03-17
Accepted: 2024-04-17
Published Online: 2024-06-20

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

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

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https://doi.org/10.1515/math-2024-0015
Keywords for this article
Schrödinger-Poisson system; variational methods; p-Laplacian; fibering method
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  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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