Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth

Quan Hai; Jing Zhang

doi:10.1515/dema-2025-0111

Article Open Access

Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth

Quan Hai and Jing Zhang

Published/Copyright: March 25, 2025

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the $p$ -biharmonic operator:

$M ∫ Ω ∣ Δ u ∣ p d x Δ p 2 u − Δ p u = λ ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q − 2 u + ∣ u ∣ p * − 2 u + f , in Ω , u = Δ u = 0 , on ∂ Ω ,$

where $Ω ⊂ R N$ , $N ≥ 3$ is a smooth bounded domain, $1 < p < N 2$ , $0 < μ < N$ , $p < 2 q < p *$ , $p * = N p N − 2 p$ denotes the Sobolev conjugate of $p$ . $M$ is a nondecreasing and continuous function, and $λ > 0$ is a parameter. $Δ p 2 u ≔ Δ ( ∣ Δ u ∣ p − 2 Δ u )$ is the operator of fourth order called the p-biharmonic operator, $Δ p u ≔ div ( ∣ ∇ u ∣ p − 2 ∇ u )$ is the $p$ -Laplacian operator. $f ≥ 0$ , $f ∈ L p p − 1 ( Ω )$ , and $∣ f ∣ p p − 1$ is sufficiently small. Using the concentration-compactness principle together with the mountain pass theorem, we obtain the existence of nontrivial solutions for the aforementioned problem in both nondegenerate and degenerate cases.

Keywords: nonhomogeneous; p-biharmonic operator; nondegenerate; degenerate

MSC 2010: 35A01; 35A15; 35B33

1 Introduction

In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard problem involving $p$ -biharmonic operator:

(1.1)

$M ∫ Ω ∣ Δ u ∣ p d x Δ p 2 u − Δ p u = λ ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q − 2 u + ∣ u ∣ p * − 2 u + f , in Ω , u = Δ u = 0 , on ∂ Ω ,$

where $Ω ⊂ R N$ , $N ≥ 3$ is a smooth bounded domain, $1 < p < N 2$ , $0 < μ < N$ , $p < 2 q < p *$ , and $p * = N p N − 2 p$ denotes the Sobolev conjugate of $p$ . $M : R 0 + ≔ [ 0 , + ∞ ) → ( 0 , + ∞ )$ is a nondecreasing and continuous function, and $λ > 0$ is a parameter. $Δ p 2 u ≔ Δ ( ∣ Δ u ∣ p − 2 Δ u )$ is the operator of fourth order called the p-biharmonic operator, $Δ p u ≔ div ( ∣ ∇ u ∣ p − 2 ∇ u )$ is the p-Laplacian operator. $f ≥ 0$ , $f ∈ L p p − 1 ( Ω )$ , and $∣ f ∣ p p − 1$ is sufficiently small.

When $f ≡ 0$ , problem (1.1) is related to the following problem of Kirchhoff type:

(1.2)

$M ∫ Ω ∣ Δ u ∣ p d x Δ p 2 u − Δ p u = g ( x , u )$

(see [1] for the meaning of the problem in physics and engineering). The problem involving $p$ -Laplacian and $p$ -biharmonic operators has been developed very quickly in recent decades; the results of problem (1.2) in the subcritical case were widely discussed; the properties of the solutions, including existence [2,3], sign-changing [4], multiplicity [5,6]; and infinitely many solutions [7], were gained. Meanwhile, the results of problem (1.2) in the critical case are also abundant. Figueiredo and Nascimento [8] obtained the existence result and used genus theory due to Krasnoselskii to show the multiplicity result. Using the concentration-compactness principle in [9], problem (1.2) had at least $m$ pairs of solutions. Other results on criticality can be found in previous works [10–12].

For the nonlinearity $g ( x , u ) = ( ∣ x ∣ − α ⁎ ∣ u ∣ q ) ∣ u ∣ q − 2 u$ , equation (1.2) is related to

(1.3)

$Δ p 2 u − Δ p u = A ∣ x ∣ α ⁎ ∣ u ∣ q ∣ u ∣ q − 2 u , x ∈ R N ,$

where $q ∈ ( 1 , N )$ . Equation (1.3) is usually called the nonlinear Choquard equation, which was introduced by Pekar [13] and described the quantum mechanics of a polaron at rest. The discussion of the Choquard equation (1.3) has always been a hot topic of concern for scholars. Up to date, equation (1.3) has achieved many excellent achievements when $p = 2$ . Chen and Chen [14] considered biharmonic Choquard equation with Hardy-Littlewood-Sobolev upper critical and combined nonlinearities, the existence of normalized ground-state solutions were first obtained, and then, a class of solutions that are not ground states and located at a mountain pass level of the energy functional were also illustrated. Multiple solutions for critical Choquard-Kirchhoff-type equations can be seen in [15] using the concentration-compactness principle and genus theory. Liu and Zhang [16] concerned with problem (1.3) with a nonlinearity $f$ on weighted lattice graphs and proved that (1.3) possesses a mountain pass solution and a ground-state solution. Pucci et al. [17] obtained existence of nonnegative solutions via the mountain pass theorem when $g$ satisfies superlinear growth conditions, and existence of nonnegative solutions was also investigated when $g$ is sublinear at infinity via the Ekeland variational principle. Using the Nehari manifold and fibering map analysis, Anu Rani and Sarika Goyal [18] took into account the critical Choquard equation involving sign-changing weight functions and demonstrated the existence of two nontrivial solutions of the problem with respect to parameter $λ$ . The 1-biharmonic Choquard equation with steep potential well was considered in [19], and the existence of a nontrivial solution $u λ$ via variational methods and the concentration behavior of $u λ$ were testified.

For more general $p$ , most of the existing results about problem (1.3) are involving $p$ -Laplacian operator. Bai et al. [20] considered critical Kirchhoff-Choquard-type equation involving the fractional $p$ -Laplacian on the Heisenberg group, and the existence of infinitely many solutions is obtained according to Krasnoselskii genus theorem. In addition, using the fractional version of the concentration-compactness principle, Bai et al. [20] also proved that the corresponding equation has $m$ pairs of solutions. Chen and Feng [21] studied a fractional $( p , q )$ -Kirchhoff Choquard problem with a critical Hardy-Sobolev term and magnetic field and obtained a nontrivial solution. Liu and Chen [22] considered a $p$ -Choquard equations with singular potential and doubly critical exponents; by applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions-type theorem, the existence of ground-state solutions was established. The existence of at least one weak solution was obtained using Nehari manifold approach for the critical double-phase Choquard problems with singular nonlinearity in [23]. In [24], a critical Choquard-Kirchhoff problem with $p ( x )$ -Laplace operator was under consideration, and the existence of nontrivial radial solutions was provided in both nondegenerate and degenerate cases.

As far as we know, there are scarce existing results about nonhomogeneous Choquard equation involving $p$ -biharmonic operator (1.1), i.e., $f ≢ 0$ . Using variational methods, Shen et al. [25] obtained the existence of multiple solutions in a smooth bounded domain of $R N$ . Multiplicity results are obtained making use of the Ekeland variational principle and the mountain pass theorem in [26] when the nonlinearity $g$ satisfied the Ambrosetti-Rabinowitz condition. Problem (1.1) involving $p$ -Laplacian operator in subcritical case was investigated in [27], and the existence of at least two nontrivial solutions was obtained using Nehari manifold and minimax methods. When $p = 2$ and $M = 0$ , critical point theorem for noneven functionals was applied in [28] and proved the existence of infinitely many virtual critical points for equation (1.1) with two classes of potential. Other related results about nonhomogeneous Choquard equation can be found in [29–31] and the references therein.

Let us assume throughout this article that for two different nondegenerate case:

there exists $m 0 > 0$ , such that $M ( t ) ≥ m 0$ , $∀ t ∈ R 0 +$ ;
there exists $σ ∈ ( p q , 1 ]$ such that $M ^ ( t ) ≥ σ M ( t ) t$ , $∀ t ∈ R 0 +$ , where $M ^ ( t ) = ∫ 0 t M ( τ ) d τ$ , and degenerate case;
there exists $m 1 > 0$ such that $M ( t ) ≥ m 1 t 1 σ − 1$ for all $t ∈ R +$ and $M ( 0 ) = 0$ .

A typical example of $M$ is given by $M ( t ) = a + b t θ − 1$ for $t ∈ R 0 +$ , where $a ∈ R 0 + , b ∈ R 0 +$ and $a + b > 0$ . When $M$ is of this type, problem (1.1) is said to be nondegenerate if $a > 0$ , while it is called degenerate if $a = 0$ . Clearly, assumptions $( M 1 )$ – $( M 2 )$ cover the nondegenerate case and $( M 2 )$ – $( M 3 )$ are automatic in the degenerate case. From a physical point of view, the fact that $M ( 0 ) = 0$ means that the base tension of the string is zero, a very realistic model.

Motivated by the contributions on $p$ -biharmonic operators with critical exponents, the recent results recovering the degenerate [32,33] and nondegenerate cases [26,34] Kirchhoff problems, we devoted to the existence of nontrivial solution of (1.1) in both degenerate and nondegenerate cases by using concentration-compactness principle to overcome the lack of compactness of the embedding $W 2 , p ( Ω ) ↪ L p * ( Ω )$ . To our best knowledge, there are no results for (1.1) in such a generality.

The main results are stated as follows:

Theorem 1.1

(Non-degenerate case) Assume that M satisfies $( M 1 )$ – $( M 2 )$ . Then, there exists $λ * > 0$ such that for any $λ ≥ λ *$ , problem (1.1) has a nontrivial solution.

Theorem 1.2

(Degenerate case) Assume that M satisfies $( M 2 )$ – $( M 3 )$ . Then, there exists $λ * > 0$ such that for any $λ ≥ λ *$ , problem (1.1) has a nontrivial solution.

2 Preliminary

In this article, we denote by

$E ≔ W 2 , p ( Ω ) ≔ { u ∈ L p ( Ω ) : ∣ D α u ∣ ∈ L p ( Ω ) , ∣ α ∣ ≤ 2 } ,$

the usual Sobolev space whose norm is given as follows:

$‖ u ‖ = ∫ Ω ∣ Δ u ∣ p d x 1 p , u ∈ E ,$

which is equivalent to the usual norm $‖ u ‖ W k , p ( Ω ) ≔ ∑ ∣ α ∣ ≤ k ∫ Ω ∣ D α u ∣ p d x 1 p$ due to [35] when $k = 2$ .

We can then be seen in [36] that $E$ is continuously and compactly embedded into the Lebesgue space $L s ( Ω )$ endowed with the norm $∣ u ∣ r = ∫ Ω ∣ u ∣ r d x 1 r$ , $1 ≤ r < p *$ . Denote by $C r > 0$ the best constant for this embedding, i.e.,

$C r ∣ u ∣ r ≤ ‖ u ‖ , ∀ u ∈ E .$

In particular, if $S$ is the best constant for the embedding $E ↪ L p * ( Ω )$ , then it is defined by the formula

(2.1)

$S = inf u ∈ E \ { 0 } ∫ Ω ∣ Δ u ∣ p d x ∫ Ω ∣ u ∣ p * d x p p * .$

It is well known that (see [37]) $η 1$ is the first eigenvalue of $p$ -Laplacian and

(2.2)

$η 1 = inf u ∈ C 0 ∞ ( Ω ) \ { 0 } ∫ Ω ∣ ∇ u ∣ p d x ∫ Ω ∣ u ∣ p d x .$

It is well known that $η 1 > 0$ plays a crucial role in the study of $p$ -Laplacian problems.

We say that a function $u ∈ E$ is a weak solution of (1.1) if for all $v ∈ E$ , we have

$M ∫ Ω ∣ Δ u ∣ p d x ∫ Ω ∣ Δ u ∣ p − 2 Δ u Δ v d x + ∫ Ω ∣ ∇ u ∣ p − 2 ∇ u ∇ v d x − λ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q − 2 u v d x − ∫ Ω ∣ u ∣ p * − 2 u v d x − ∫ Ω f v d x = 0 .$

The energy functional corresponding to problem (1.1) is

$I λ ( u ) = 1 p M ^ ∫ Ω ∣ Δ u ∣ p d x + 1 p ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u ∣ p * d x − ∫ Ω f u d x .$

We can see that $I λ ∈ C 1 ( E , R )$ and the critical points of $I λ$ are the weak solutions of problem (1.1) that

$⟨ I λ ′ ( u ) , v ⟩ = M ∫ Ω ∣ Δ u ∣ p d x ∫ Ω ∣ Δ u ∣ p − 2 Δ u Δ v d x + ∫ Ω ∣ ∇ u ∣ p − 2 ∇ u ∇ v d x − λ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q − 2 u v d x − ∫ Ω ∣ u ∣ p * − 2 u v d x − ∫ Ω f v d x ,$

for $∀ u , v ∈ E$ .

To achieve our main result, we recall the well-known Hardy-Littlewood-Sobolev inequality from [38].

Lemma 2.1

Let $r , s > 1$ , and $0 < α < N$ be such that $1 s + 1 t − α N = 1$ . Let $g ∈ L s ( R N )$ and $h ∈ L t ( R N )$ , there exists a sharp constant $C ( s , t , N , α )$ , independent of $g$ and $h$ , such that

$∫ R N ( ∣ x ∣ − α ⁎ g ) h d x ≤ C ( s , t , N , α ) ∣ g ∣ s ∣ h ∣ t .$

3 Non-degenerate case

In this section, we assume that $( M 1 )$ – $( M 2 )$ are satisfied, and we begin the study of the nondegenerate case of (1.1). Lemma 3.1 is crucial to prove the main results for problem (1.1), i.e., $I λ$ satisfies the (PS) $λ$ condition.

Lemma 3.1

Let ${ u n } ⊂ E$ be a $( P S ) c λ$ sequence of functional $I λ$ , i.e.,

(3.1)

$I λ ( u n ) → c λ , I λ ′ ( u n ) → 0 ,$

for some $c λ ∈ R$ as $n → ∞$ . If

$c λ < σ p − 1 p * ( m 0 S ) N 2 p − C ˜ ∣ f ∣ r r ,$

where $1 p + 1 r = 1$ , then there exists a subsequence of ${ u n }$ strongly convergent in E.

Proof

We first show that ${ u n }$ is bounded in $E$ . Indeed, note that by relation (3.1), conditions $( M 1 )$ and $( M 2 )$ , we have

By the Hölder and Young inequalities that for small enough $ε$ ,

(3.2)

$∫ Ω f u d x ≤ ∣ f ∣ r ∣ u ∣ p ≤ ε ∣ u ∣ p p + C ε ∣ f ∣ r r ,$

where $1 p + 1 r = 1$ . Then, by (2.2), let $ε = η 1 1 p − 1 2 q 2 1 − 1 2 q$ ,

$c λ + o n ( 1 ) ‖ u n ‖ ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x − 1 − 1 2 q ε ∣ u ∣ p p − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x + η 1 2 1 p − 1 2 q ∣ u ∣ p p − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x − 1 − 1 2 q C ε ∣ f ∣ r r .$

Furthermore, we obtain

(3.3)

$‖ u n ‖ p ≤ c 1 + o n ( 1 ) ‖ u n ‖ ,$

where $o n ( 1 ) → 0$ and $c 1 > 0$ . Thus, (3.3) implies that ${ u n }$ is bounded in $E$ . Hence, up to a subsequence, we may assume that

$u n ⇀ u in E , u n → u in L s ( Ω ) , 1 ≤ s < p * , u n ( x ) → u ( x ) a.e. x ∈ Ω .$

Using the concentration-compactness principle due to [39,40], there exist two bounded nonnegative measures $μ$ and $ν$ on $R N$ and some at least countable family of points $( x j ) j ∈ Λ ⊂ Ω ¯$ and of nonnegative numbers $( μ j ) j ∈ Λ$ and $( ν j ) j ∈ Λ$ such that

(3.4)

$∣ u n ∣ p * ⇀ ν = ∣ u ∣ p * + ∑ j ∈ Λ ν j δ x j ,$

(3.5)

$∣ Δ u n ∣ p ⇀ μ ≥ ∣ Δ u ∣ p + ∑ j ∈ Λ μ j δ x j ,$

(3.6)

$S ν j p p * ≤ μ j ,$

for all $j ∈ Λ$ , where $δ x j$ is the Dirac mass at $x j ∈ Ω ¯$ , where $S$ is the best constant of the embedding $E ↪ L p * ( Ω )$ , given by (2.1). For each $j ∈ Λ$ and $ε > 0$ , let us consider $ϕ j , ε ∈ C ∞ ( R N )$ such that

$ϕ j , ε ≡ 1 in B ε ( x j ) , ϕ j , ε ≡ 0 on Ω \ B 2 ε ( x j ) , ∣ ∇ ϕ j , ε ∣ ∞ ≤ 2 ε , ∣ Δ ϕ j , ε ∣ ≤ 2 ε 2 ,$

where $x j ∈ Ω ¯$ belongs to the support of $ν$ . Since ${ u n ϕ j , ε }$ is bounded in the space $E$ , it then follows from (3.1) that $⟨ I λ ′ ( u n ) , u n ϕ j , ε ⟩ → 0$ as $n → ∞$ , i.e.,

(3.7)

$⟨ I λ ′ ( u n ) , u n ϕ j , ε ⟩ = M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n Δ ( u n ϕ j , ε ) d x + ∫ Ω ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n ϕ j , ε ) d x − λ ∫ Ω ( ∣ x ∣ − μ * ∣ u n ∣ q ) ∣ u n ∣ q − 2 u n ( u n ϕ j , ε ) d x − ∫ Ω ∣ u n ∣ p * − 2 u n ( u n ϕ j , ε ) d x − ∫ Ω f ⋅ ( u n ϕ j , ε ) d x → 0 ,$

as $n → ∞$ . It is noted that $Δ ( u n ϕ j , ε ) = Δ u n ⋅ ϕ j , ε + 2 ∇ u n ⋅ ∇ ϕ j , ε + u n ⋅ Δ ϕ j , ε$ [41], then (3.7) gives us

(3.8)

$M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ϕ j , ε ∣ Δ u n ∣ p d x − ∫ Ω ϕ j , ε ∣ u n ∣ p * d x = λ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u n ∣ q ) ∣ u n ∣ q − 2 u n ( u n ϕ j , ε ) d x + ∫ Ω f ⋅ ( u n ϕ j , ε ) d x − 2 M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n ( ∇ u n ⋅ ∇ ϕ j , ε ) d x − M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n ( u n ⋅ Δ ϕ j , ε ) d x − ∫ Ω ϕ j , ε ∣ ∇ u n ∣ p d x − ∫ Ω ∣ ∇ u n ∣ p − 2 u n ( ∇ u n ⋅ ∇ ϕ j , ε ) d x → 0 .$

By Lemma 2.1 and the Lebesgue-dominated convergence theorem, we have

(3.9)

$limsup ε → 0 limsup n → ∞ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u n ∣ q ) ∣ u n ∣ q − 2 u n ( u n ϕ j , ε ) d x = limsup ε → 0 ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q ϕ j , ε d x = 0 .$

Using the Hölder inequality and the boundedness of the sequence ${ u n }$ in $E$ , then

(3.10)

$∫ Ω f ⋅ ( u n ϕ j , ε ) d x ≤ ∫ B 2 ε ( x j ) ∩ Ω ∣ f ∣ r d x 1 r ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p ∣ ϕ j , ε ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p ∣ ϕ j , ε ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p * d x 1 p * ∫ B 2 ε ( x j ) ∩ Ω ∣ ϕ j , ε ∣ 2 N d x 2 N ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p * d x 1 p * → 0 ,$

as $n → ∞$ and $ε → 0$ .

Using the Hölder inequality and the boundedness of the sequence ${ u n }$ in $E$ again, then

(3.11)

$∫ Ω ∣ Δ u n ∣ p − 2 Δ u n ( ∇ u n ⋅ ∇ ϕ j , ε ) d x ≤ ∫ B 2 ε ( x j ) ∩ Ω ∣ Δ u n ∣ p d x p − 1 p ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ p ∣ ∇ ϕ j , ε ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ p ∣ ∇ ϕ j , ε ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ N p N − p d x N − p N p ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ ϕ j , ε ∣ N d x 1 N ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ N p N − p d x N − p N p → 0 ,$

as $n → ∞$ and $ε → 0$ . Since ${ u n }$ is bounded in $E$ , we may assume that $∫ Ω ∣ Δ u n ∣ p d x → τ ≥ 0$ as $n → ∞$ . Observing that $M ( t )$ is continuous, we then have

$M ∫ Ω ∣ Δ u n ∣ p d x → M ( τ ) ≥ m 0 > 0 ,$

as $n → ∞$ . Then, by (3.11),

(3.12)

$M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n ( ∇ u n ⋅ ∇ ϕ j , ε ) d x → 0 ,$

as $n → ∞$ and $ε → 0$ .

Similarly,

$∫ Ω ∣ Δ u n ∣ p − 2 Δ u n ( u n ⋅ Δ ϕ j , ε ) d x$

$≤ ∫ B 2 ε ( x j ) ∩ Ω ∣ Δ u n ∣ p d x p − 1 p ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p ∣ Δ ϕ j , ε ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p ∣ Δ ϕ j , ε ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p * d x 1 p * ∫ B 2 ε ( x j ) ∩ Ω ∣ Δ ϕ j , ε ∣ 2 N d x 2 N ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p * d x 1 p * → 0 ,$

as $n → ∞$ and $ε → 0$ . So that

(3.13)

$M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n ( u n ⋅ Δ ϕ j , ε ) d x → 0 ,$

as $n → ∞$ and $ε → 0$ . Furthermore,

(3.14)

$∫ Ω ϕ j , ε ∣ ∇ u n ∣ p d x ≤ ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ N p N − p d x N − p N ∫ B 2 ε ( x j ) ∩ Ω ∣ ϕ j , ε ∣ N p d x p N ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ N p N − p d x N − p N → 0 ,$

as $n → ∞$ and $ε → 0$ . Moreover,

(3.15)

$∫ Ω ∣ ∇ u n ∣ p − 2 u n ( ∇ u n ⋅ ∇ ϕ j , ε ) d x ≤ ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ p ∣ ∇ ϕ j , ε ∣ p p − 1 d x p − 1 p ∫ B 2 ε ( x j ) ∩ Ω ∣ u n ∣ p d x 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ p ∣ ∇ ϕ j , ε ∣ p p − 1 d x p − 1 p ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ N p N − p d x ( N − p ) ( p − 1 ) N p ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ ϕ j , ε ∣ N p − 1 d x p − 1 N ≤ C ∫ B 2 ε ( x j ) ∩ Ω ∣ ∇ u n ∣ N p N − p d x ( N − p ) ( p − 1 ) N p → 0 ,$

as $n → ∞$ and $ε → 0$ . Taking $n → ∞$ and then as $ε → 0$ in (3.8), and using (3.9)–(3.15), we arrive at

$∫ Ω M ( τ ) ϕ j , ε d μ = ∫ Ω ϕ j , ε d ν .$

Letting $ε → 0$ , we conclude that $M ( τ ) μ j = ν j$ . By $( M 1 )$ , $m 0 μ j ≤ ν j$ . From this and (3.6), we obtain

(3.16)

$m 0 S ν j p p * ≤ m 0 μ j ≤ ν j ,$

which implies that $ν j = 0$ or $ν j ≥ ( m 0 S ) N 2 p , j ∈ Λ$ .

From conditions $( M 1 )$ , $( M 2 )$ , (2.2), and (3.2), let $ε = η 1 1 p − 1 2 q 2 1 − 1 2 q$ , and we have

$c λ + o n ( 1 ) ‖ u n ‖ = I λ ( u n ) − 1 2 q ⟨ I λ ′ ( u n ) , u n ⟩ = 1 p M ^ ∫ Ω ∣ Δ u n ∣ p d x + 1 p ∫ Ω ∣ ∇ u n ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u n ∣ p * d x − ∫ Ω f u d x − 1 2 q M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p d x + 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 2 q ∫ Ω ∣ u n ∣ p * d x − 1 2 q ∫ Ω f u d x = 1 p M ^ ∫ Ω ∣ Δ u n ∣ p d x − 1 2 q M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p d x + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q ∫ Ω f u d x ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q ∫ Ω f u d x ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q ε ∣ u ∣ p p − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x + η 1 2 1 p − 1 2 q ∣ u ∣ p p + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 0 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q C ε ∣ f ∣ r r ,$

and then, we obtain

(3.17)

$c λ ≥ m 0 σ p − 1 2 q lim n → ∞ ∫ Ω ∣ Δ u n ∣ p d x + 1 2 q − 1 p * lim n → ∞ ∫ Ω ∣ u n ∣ p * d x − C ˜ ∣ f ∣ r r .$

Using (3.4) and (3.5), it implies that

$lim n → ∞ ∫ Ω ∣ u n ∣ p * d x = ∫ Ω ∣ u ∣ p * d x + ∑ j ∈ Λ ν j ≥ ν j , ∀ j ∈ Λ ,$

and

$lim n → ∞ ∫ Ω ∣ Δ u n ∣ p d x = ∫ Ω ∣ Δ u ∣ p d x + ∑ j ∈ Λ μ j ≥ μ j , ∀ j ∈ Λ ,$

if $ν s > 0$ and $μ s > 0$ for some $s ∈ Λ$ , we deduce from relations (3.16) and (3.17) that

$c λ ≥ σ p − 1 p * ( m 0 S ) N 2 p − C ˜ ∣ f ∣ r r ,$

which is in contradiction with the range of $c λ$ . This leads to the fact that $μ j = 0$ for any $j ∈ Λ$ . Thus, $ν j = 0$ for any $j ∈ Λ$ and

$lim n → ∞ ∫ Ω ∣ u n ∣ p * = ∫ Ω ∣ u ∣ p * d x ,$

and by the Brezis-Lieb lemma [42], the sequence ${ u n }$ converges strongly to $u$ in $L p * ( Ω )$ . For this reason, by the Hölder inequality, we deduce that

(3.18)

$∫ Ω ∣ u n ∣ p * − 2 u n ( u n − u ) d x ≤ ∫ Ω ∣ u n ∣ p * p * − 1 p * ∫ Ω ∣ u n − u ∣ p * 1 p * → 0 ,$

as $n → ∞$ . Also, by Lemma 2.1 for $s = 2 N N + μ$ ,

(3.19)

$∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u n ∣ q ) ∣ u n ∣ q − 2 u n ( u n − u ) d x ≤ C ∫ Ω ∣ u n ∣ q s 1 s ∫ Ω ∣ u n ∣ ( q − 1 ) s ∣ u n − u ∣ s 1 s ≤ C ∫ Ω ∣ u n ∣ q s q − 1 q ∫ Ω ∣ u n − u ∣ q s 1 q 1 s → 0 ,$

as $n → ∞$ . Moreover,

(3.20)

$∫ Ω f ( u n − u ) d x ≤ ∫ Ω ∣ f ∣ r 1 r ∫ Ω ∣ u n − u ∣ p 1 p → 0 ,$

as $n → ∞$ .

Since the sequence ${ u n }$ converges weakly to $u$ in $E$ , the sequence ${ u n − u }$ is bounded in $E$ and $⟨ I λ ′ ( u n ) , u n − u ⟩ → 0$ , as $n → ∞$ , i.e.,

(3.21)

$⟨ I λ ′ ( u n ) , u n − u ⟩ = M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n Δ ( u n − u ) d x + ∫ Ω ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n − u ) d x − λ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u n ∣ q ) ∣ u n ∣ q − 2 u n ( u n − u ) d x − ∫ Ω ∣ u n ∣ p * − 2 u n ( u n − u ) d x − ∫ Ω f ( u n − u ) d x → 0 .$

From relations (3.18)–(3.21), we have

$M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n Δ ( u n − u ) d x + ∫ Ω ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n − u ) d x → 0 ,$

and by $( M 1 )$ , it follows that

$lim n → ∞ ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n Δ ( u n − u ) d x = 0$

and

$lim n → ∞ ∫ Ω ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n − u ) d x = 0 .$

Hence, we derive $u n → u$ in $E$ via a standard argument (see, e.g., [43], Proof of Lemma 3.4). The proof is complete.□

Now we state the general version of the mountain pass theorem in [44], which will be used later.

Theorem 3.1

Let $I λ$ be a functional on a Banach space X and $I λ ∈ C 1 ( X , R )$ . Let us assume that there exists $α , ρ > 0$ such that

$I λ ( u ) ≥ α$ , $∀ u ∈ X$ with $‖ u ‖ = ρ$ ,
$I λ ( 0 ) = 0$ and $I λ ( e ) < α$ for some $e ∈ X$ with $‖ e ‖ > ρ$ .

Let us define

$Γ = { γ ∈ C ( [ 0,1 ] , X ) : γ ( 0 ) = 0 , γ ( 1 ) = e }$ , and

$c λ = inf γ ∈ Γ max t ∈ [ 0,1 ] I λ ( γ ( t ) ) .$

Then, there exists a sequence ${ u n } ⊂ X$ such that $I λ ( u n ) → c λ$ and $I λ ′ ( u n ) → 0$ in $X ′$ (dual of X).

In the following, we show that $I λ$ satisfies geometric properties (i) and (ii) of mountain pass.

Lemma 3.2

The functional $I λ$ , satisfies assumptions (i) and (ii) in Theorem 3.1.

Proof

Let $ε = η 1 2 p$ , and according to $( M 1 ) , ( M 2 )$ , (2.2), (3.2), and Lemma 2.1,

$I λ ( u ) = 1 p M ^ ∫ Ω ∣ Δ u ∣ p d x + 1 p ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u ∣ p * d x − ∫ Ω f u d x ≥ σ p M ∫ Ω ∣ Δ u ∣ p d x ∫ Ω ∣ Δ u ∣ p d x + 1 p ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u ∣ p * d x − ε ∫ Ω ∣ u ∣ p d x − C ε ∣ f ∣ r r ≥ σ m 0 p ∫ Ω ∣ Δ u ∣ p d x + 1 p − ε η 1 ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q C ∫ Ω ∣ u ∣ q s d x 2 s − 1 p * ∫ Ω ∣ u ∣ p * d x − C ε ∣ f ∣ r r ≥ σ m 0 p ‖ u ‖ p − λ C 2 q C q s 2 q ‖ u ‖ 2 q − 1 p * S p * p ‖ u ‖ p * − C ε ∣ f ∣ r r ,$

where $s = 2 N N + α$ , so that for sufficiently small $∣ f ∣ r$ , there exists small enough $ρ > 0$ and $α > 0$ such that $I λ ( u ) ≥ α$ for every $‖ u ‖ = ρ$ and all $λ > 0$ . Then, assumptions (i) in Theorem 3.1 is satisfied.

(ii) First of all, by condition $( M 2 )$ , we obtain that

(3.22)

$M ^ ( t ) ≤ M ^ ( t 0 ) t 0 1 σ t 1 σ = C ^ t 1 σ , ∀ t ≥ t 0 > 0 .$

From (3.22) and $σ ∈ ( p 2 q , 1 ]$ ,

$I λ ( t u ) ≤ C ^ p t p σ ∫ Ω ∣ Δ u ∣ p d x 1 σ + t p p ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q t 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − t p * p * ∫ Ω ∣ u ∣ p * d x − t ∫ Ω f u d x ,$

and hence, $I λ ( t u ) → − ∞$ as $t → + ∞$ . Therefore, there exists $t 0$ large enough such that $I λ ( t 0 u ) < 0$ . Then, we take $e = t 0 u$ and $I λ ( e ) < 0$ . Hence, (ii) of Theorem 3.1 holds true.□

Proof of Theorem 1.1

We claim that

(3.23)

$c λ = inf γ ∈ Γ max t ∈ [ 0,1 ] I λ ( γ ( t ) ) < σ p − 1 p * ( m 0 S ) N 2 p − C ˜ ∣ f ∣ r r .$

Now we assume (3.23) holds true; then, Lemmas 3.1–3.2 and Theorem 3.1 give the existence of nontrivial radial critical points of $I λ$ .

To prove (3.23), we choose $v 0 ∈ E$ such that

$‖ v 0 ‖ = 1 , ∣ v 0 ∣ p * > 0 , lim t → ∞ I λ ( t v 0 ) = − ∞ ,$

then $sup t ≥ 0 I λ ( t v 0 ) = I λ ( t λ v 0 )$ for some $t λ > 0$ . Hence, $t λ$ satisfies

(3.24)

$t λ p M ( t λ p ) + t λ p ∫ Ω ∣ ∇ v 0 ∣ p d x = λ t λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ v 0 ∣ q ) ∣ v 0 ∣ q d x + t λ p * ∫ Ω ∣ v 0 ∣ p * d x + t λ ∫ Ω f v 0 d x .$

Then, by (3.2),

(3.25)

$λ t λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ v 0 ∣ q ) ∣ v 0 ∣ q d x + t λ p * ∫ Ω ∣ v 0 ∣ p * d x − t λ ∣ f ∣ r ∣ v 0 ∣ p ≤ t λ p M ( t λ p ) + t λ p ∫ Ω ∣ ∇ v 0 ∣ p d x ≤ λ t λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ v 0 ∣ q ) ∣ v 0 ∣ q d x + t λ p * ∫ Ω ∣ v 0 ∣ p * d x + t λ ∣ f ∣ r ∣ v 0 ∣ p ,$

so because of the previous inequality in (3.25) that

(3.26)

$t λ p M ( t λ p ) + t λ p ∫ Ω ∣ ∇ v 0 ∣ p d x + t λ ∣ f ∣ r ∣ v 0 ∣ p ≥ t λ p * ∫ Ω ∣ v 0 ∣ p * d x .$

Then, by $( M 2 )$ , (3.22), and (3.26),

(3.27)

$C 1 t λ p σ + t λ p ∫ Ω ∣ ∇ v 0 ∣ p d x + t λ ∣ f ∣ r ∣ v 0 ∣ p ≥ 1 σ M ^ ( t λ p ) + t λ p ∫ Ω ∣ ∇ v 0 ∣ p d x + t λ ∣ f ∣ r ∣ v 0 ∣ p ≥ t λ p * ∫ Ω ∣ v 0 ∣ p * d x ,$

then (3.27) shows that ${ t λ } λ$ is bounded.

We claim that $t λ → 0$ as $λ → ∞$ . Arguing by contradiction, we can assume that there exist $t 0 > 0$ and a sequence $λ n$ with $λ n → ∞$ as $n → ∞$ such that $t λ n → t 0$ as $n → ∞$ . By Lemma 2.1 and Lebesgue-dominated convergence theorem, we deduce

$∫ Ω ( ∣ x ∣ − μ ⁎ ∣ t λ n v 0 ∣ q ) ∣ t λ n v 0 ∣ q d x → ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ t 0 v 0 ∣ q ) ∣ t 0 v 0 ∣ q d x ,$

as $n → ∞$ , so that

$λ n ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ t 0 v 0 ∣ q ) ∣ t 0 v 0 ∣ q d x → ∞ ,$

as $n → ∞$ . Hence, (3.25) implies that

$t 0 p M ( t 0 p ) + t 0 p ∫ Ω ∣ ∇ v 0 ∣ p d x = ∞ ,$

which is absurd. Therefore, $t λ → 0$ as $λ → ∞$ . Furthermore, we deduce from (3.24) that

$lim λ → ∞ λ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ t 0 v 0 ∣ q ) ∣ t 0 v 0 ∣ q d x → 0 .$

From this, $t λ → 0$ as $λ → ∞$ and the definition of $I λ$ , we obtain

$lim λ → ∞ sup t ≥ 0 I λ ( t v 0 ) = lim λ → ∞ I λ ( t λ v 0 ) = 0 .$

Then, there exists $λ * > 0$ such that for any $λ ≥ λ *$ ,

$sup t ≥ 0 I λ ( t v 0 ) < σ p − 1 p * ( m 0 S ) N 2 p − C ˜ ∣ f ∣ r r ,$

for sufficiently small $∣ f ∣ r$ . If we take $e = T v 0$ , with $T$ large enough to verify $I λ ( e ) < 0$ , then we obtain $c λ ≤ max t ∈ [ 0,1 ] I λ ( γ ( t ) )$ by taking $γ ( t ) = t T v 0$ . Therefore, $c λ ≤ sup t ≥ 0 I λ ( t v 0 ) < σ p − 1 p * ( m 0 S ) N 2 p − C ˜ ∣ f ∣ r r$ for $λ$ large enough and $∣ f ∣ r$ small enough.□

4 Degenerate case

In this section, we assume that $( M 2 )$ and $( M 3 )$ are satisfied and start with the study of the degenerate case of (1.1). We first give a crucial Lemma 4.1 in the proof of main results for problem (1.1), which means that $I λ$ is said to satisfy the $( P S ) c λ$ condition.

Lemma 4.1

Let ${ u n } ⊂ E$ be a $( P S ) c λ$ sequence of functional $I λ$ , i.e.,

(4.1)

$I λ ( u n ) → c λ , I λ ′ ( u n ) → 0$

for some $c λ ∈ R$ as $n → ∞$ . If

$c λ < σ p − 1 p * m 1 S 1 σ p * σ p * σ − p − C ˜ ∣ f ∣ r r ,$

where $1 p + 1 r = 1$ , then there exists a subsequence of ${ u n }$ strongly convergent in E.

Proof

If $inf n ≥ 1 ‖ u n ‖ = 0$ , then there exists a subsequence of ${ u n }$ still denoted by ${ u n }$ such that $u n → 0$ in $E$ as $n → ∞$ . Thus, we assume that $d ≔ inf n ≥ 1 ‖ u n ‖ > 0$ in the following proof.

By (4.1), $I λ ( u n ) → c λ$ , and $I λ ′ ( u n ) → 0$ , then by conditions $( M 2 )$ and $( M 3 )$ ,

Then, by (2.2) and (3.2), let $ε = η 1 1 p − 1 2 q 2 1 − 1 2 q$ ,

$c λ + o n ( 1 ) ‖ u n ‖ ≥ m 1 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x 1 σ + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x − 1 − 1 2 q ε ∣ u ∣ p p − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 1 σ p − 1 2 q ‖ u n ‖ p σ + η 1 2 1 p − 1 2 q ∣ u ∣ p p − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 1 σ p − 1 2 q ‖ u n ‖ p σ − 1 − 1 2 q C ε ∣ f ∣ r r .$

Furthermore, we obtain

(4.2)

$‖ u n ‖ p σ ≤ c 2 + o n ( 1 ) ‖ u n ‖ ,$

where $o n ( 1 ) → 0$ and $c 2 > 0$ . For $1 < p ≤ p σ$ , then (4.2) implies that ${ u n }$ is bounded in $E$ . Hence, up to a subsequence, we may assume that

$u n ⇀ u , in E ,$

$u n → u , in L s ( Ω ) , 1 ≤ s < p * ,$

$u n ( x ) → u ( x ) , a.e. x ∈ Ω .$

(4.3)

$∣ u n ∣ p * ⇀ ν = ∣ u ∣ p * + ∑ j ∈ Λ ν j δ x j ,$

(4.4)

$∣ Δ u n ∣ p ⇀ μ ≥ ∣ Δ u ∣ p + ∑ j ∈ Λ μ j δ x j ,$

for all $j ∈ Λ$ , where $δ x j$ is the Dirac mass at $x j ∈ Ω ¯$ .

Now we claim that

(4.5)

$u n → u , in E ,$

as $n → ∞$ . For each $j ∈ Λ$ and $ε > 0$ , let us consider $ϕ j , ε ∈ C ∞ ( R N )$ such that

$ϕ j , ε ≡ 1 in B ε ( x j ) , ϕ j , ε ≡ 0 , on Ω \ B 2 ε ( x j ) , ∣ ∇ ϕ j , ε ∣ ∞ ≤ 2 ε , ∣ Δ ϕ j , ε ∣ ≤ 2 ε 2 ,$

where $x j ∈ Ω$ belongs to the support of $ν$ . Since ${ u n ϕ j , ε }$ is bounded in the space $E$ , it then follows from (4.1) that $⟨ I λ ′ ( u n ) , u n ϕ j , ε ⟩ → 0$ as $n → ∞$ , i.e.,

(4.6)

$⟨ I λ ′ ( u n ) , u n ϕ j , ε ⟩ = M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p − 2 Δ u n Δ ( u n ϕ j , ε ) d x + ∫ Ω ∣ ∇ u n ∣ p − 2 ∇ u n ∇ ( u n ϕ j , ε ) d x − λ ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u n ∣ q ) ∣ u n ∣ q − 2 u n ( u n ϕ j , ε ) d x − ∫ Ω ∣ u n ∣ p * − 2 u n ( u n ϕ j , ε ) d x − ∫ Ω f ⋅ ( u n ϕ j , ε ) d x → 0 ,$

as $n → ∞$ . It is noted that $Δ ( u n ϕ j , ε ) = Δ u n ⋅ ϕ j , ε + 2 ∇ u n ⋅ ∇ ϕ j , ε + u n ⋅ Δ ϕ j , ε$ [41], then (4.6) gives us

(4.7)

Note that by $( M 3 )$ , there holds

$lim ε → 0 sup n → ∞ M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ϕ j , ε ∣ Δ u n ∣ p d x ≥ m 1 ∫ Ω ϕ j , ε ∣ Δ u n ∣ p d x 1 σ .$

Using a similar discussion as in Lemma 3.1, we deduce that

(4.8)

$lim ε → 0 sup n → ∞ M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ϕ j , ε ∣ Δ u n ∣ p d x ≥ m 1 μ j 1 σ$

and

(4.9)

$lim ε → 0 sup n → ∞ ∫ Ω ϕ j , ε ∣ u n ∣ p * d x = ν j .$

Like the proof in Lemma 3.1, substituting (4.8)–(4.9) into (4.7), we obtain

$m 1 μ j 1 σ ≤ ν j .$

It follows from $S ν j p p * ≤ μ j$ for all $j ∈ Λ$ that $μ j = 0$ or

(4.10)

$ν j ≥ m 1 S 1 σ p * σ p * σ − p .$

From conditions $( M 2 )$ , $( M 3 )$ , (2.2), and (3.2), let $ε = η 1 1 p − 1 2 q 2 1 − 1 2 q$ , and we have

$c λ + o n ( 1 ) ‖ u n ‖ = I λ ( u n ) − 1 2 q ⟨ I λ ′ ( u n ) , u n ⟩ = 1 p M ^ ∫ Ω ∣ Δ u n ∣ p d x + 1 p ∫ Ω ∣ ∇ u n ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u n ∣ p * d x − ∫ Ω f u d x − 1 2 q M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p d x + 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 2 q ∫ Ω ∣ u n ∣ p * d x − 1 2 q ∫ Ω f u d x = 1 p M ^ ∫ Ω ∣ Δ u n ∣ p d x − 1 2 q M ∫ Ω ∣ Δ u n ∣ p d x ∫ Ω ∣ Δ u n ∣ p d x + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q ∫ Ω f u d x ≥ m 1 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x 1 σ + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q ∫ Ω f u d x ≥ m 1 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x 1 σ + 1 p − 1 2 q ∫ Ω ∣ ∇ u n ∣ p d x + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q ε ∣ u ∣ p p − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 1 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x 1 σ + η 1 2 1 p − 1 2 q ∣ u ∣ p p + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q C ε ∣ f ∣ r r ≥ m 1 σ p − 1 2 q ∫ Ω ∣ Δ u n ∣ p d x 1 σ + 1 2 q − 1 p * ∫ Ω ∣ u n ∣ p * d x − 1 − 1 2 q C ε ∣ f ∣ r r ,$

and then, we obtain

(4.11)

$c λ ≥ m 1 σ p − 1 2 q lim n → ∞ ∫ Ω ∣ Δ u n ∣ p d x 1 σ + 1 2 q − 1 p * lim n → ∞ ∫ Ω ∣ u n ∣ p * d x − C ˜ ∣ f ∣ r r .$

Using (4.3) and (4.4), it implies that

$lim n → ∞ ∫ Ω ∣ u n ∣ p * d x = ∫ Ω ∣ u ∣ p * d x + ∑ j ∈ Λ ν j ≥ ν j , ∀ j ∈ Λ ,$

and

$lim n → ∞ ∫ Ω ∣ Δ u n ∣ p d x = ∫ Ω ∣ Δ u ∣ p d x + ∑ j ∈ Λ μ j ≥ μ j , ∀ j ∈ Λ .$

If $ν s > 0$ and $μ s > 0$ for some $s ∈ Λ$ , we deduce from relations (4.10) and (4.11) that

$c λ ≥ σ p − 1 p * m 1 S 1 σ p * σ p * σ − p − C ˜ ∣ f ∣ r r ,$

which is in contradiction with the range of $c λ$ . This leads to the fact that $μ j = 0$ for any $j ∈ Λ$ . Thus, $ν j = 0$ for any $j ∈ Λ$ and

(4.12)

$lim n → ∞ ∫ Ω ∣ u n ∣ p * = ∫ Ω ∣ u ∣ p * d x ,$

and by the Brezis-Lieb lemma [42], the sequence ${ u n }$ converges strongly to u in $L p * ( Ω )$ from (4.12). A similar discussion as in Lemma 3.1 yields that (4.5) holds, and then, $u n → u$ in $E$ as $n → ∞$ .□

In what follows, we prove that $I λ$ satisfies geometric properties (i) and (ii) of mountain pass.

Lemma 4.2

The functional $I λ$ satisfies assumptions (i) and (ii) in Theorem 3.1.

Proof

Let $ε = η 1 2 p$ , and according to $( M 2 ) , ( M 3 )$ , (2.2), (3.2), and Lemma 2.1,

$I λ ( u ) = 1 p M ^ ∫ Ω ∣ Δ u ∣ p d x + 1 p ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u ∣ p * d x − ∫ Ω f u d x ≥ σ p M ∫ Ω ∣ Δ u ∣ p d x ∫ Ω ∣ Δ u ∣ p d x + 1 p ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q ∫ Ω ( ∣ x ∣ − μ ⁎ ∣ u ∣ q ) ∣ u ∣ q d x − 1 p * ∫ Ω ∣ u ∣ p * d x − ε ∫ Ω ∣ u ∣ p d x − C ε ∣ f ∣ r r ≥ σ m 1 p ∫ Ω ∣ Δ u ∣ p d x 1 σ + 1 p − ε η 1 ∫ Ω ∣ ∇ u ∣ p d x − λ 2 q C ∫ Ω ∣ u ∣ q s d x 2 s − 1 p * ∫ Ω ∣ u ∣ p * d x − C ε ∣ f ∣ r r ≥ σ m 1 p ‖ u ‖ p σ − λ C 2 q C q s 2 q ‖ u ‖ 2 q − 1 p * S p * p ‖ u ‖ p * − C ε ∣ f ∣ r r .$

For $p ≤ p σ < 2 q$ , as the proof in Lemma 3.2 that for sufficiently small $∣ f ∣ r$ , there exist small enough $ρ 1 > 0$ and $α 1 > 0$ such that $I λ ( u ) ≥ α 1 > 0$ for all $‖ u ‖ = ρ 1$ , and all $λ > 0$ . Hence, (i) in Theorem 3.1 holds true. Similar to Lemma 3.2, we can show that (ii) in Theorem 3.1 holds true.□

Proof of Theorem 1.2

Using the same discussion as in the proof of Theorem 1.1, we deduce that

$c λ = inf γ ∈ Γ max t ∈ [ 0,1 ] I λ ( γ ( t ) ) < σ p − 1 p * m 1 S 1 σ p * σ p * σ − p − C ˜ ∣ f ∣ r r .$

The rest of the proof is similar to Theorem 1.1.□

Funding information: Quan Hai was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (No. 2023LHMS01005), Mathematics First-class Disciplines Cultivation Fund of Inner Mongolia Normal University (No. 2024YLKY15) and the Fundamental Research Funds for the Inner Mongolia Normal University (No. 2022JBQN073). Jing Zhang was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (No. 2022MS01001), Research Program of Sscience and Technology at Universities of Inner Mongolia Autonomous Region (No. NJYT23100), and the Fundamental Research Funds for the Inner Mongolia Normal University (No. 2022JBQN072).
Author contributions: The authors declare that they have equal contributions.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Not applicable.

References

[1] J. M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61–90, DOI: https://doi.org/10.1016/0022-247X(73)90121-2. Search in Google Scholar

[2] J. Sun and T. F. Wu, Existence of nontrivial solutions for a biharmonic equation with p-Laplacian and singular sign-changing potential, Appl. Math. Lett. 66 (2017), 61–67, DOI: https://doi.org/10.1016/j.aml.2016.11.001. Search in Google Scholar

[3] J. Sun and T. F. Wu, The Nehari manifold of biharmonic equations with p-Laplacian and singular potential, Appl. Math. Lett. 88 (2019), 156–163, DOI: https://doi.org/10.1016/j.aml.2018.08.025. Search in Google Scholar

[4] W. Zhang, X. Tang, B. Cheng, and J. Zhang, Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type, Comm. Pure Appl. Anal. 15 (2016), 2161–2177, DOI: https://doi.org/10.3934/cpaa.2016032. Search in Google Scholar

[5] M. Ferrara, S. Khademloo, and S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems, Appl. Math. Comput. 234 (2014), 316–325, DOI: https://doi.org/10.1016/j.amc.2014.02.041. Search in Google Scholar

[6] J. Sun, J Chu, and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian, J. Differential Equations 262 (2017), 945–977, DOI: https://doi.org/10.1016/j.jde.2016.10.001. Search in Google Scholar

[7] D. L. Wu and F. Li, Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in RN, Comput. Math. Appl. 79 (2020), 489–499, DOI: https://doi.org/10.1016/j.camwa.2019.07.007. Search in Google Scholar

[8] G. M. Figueiredo and R. G. Nascimento, Multiplicity of solutions for equations involving a nonlocal term and the biharmonic operator, Electron. J. Differential Equations 217 (2016), 1–15. Search in Google Scholar

[9] S. Liang and Z. Zhang, Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in RN, J. Math. Phys. 57 (2016), 111505, DOI: https://doi.org/10.1063/1.4967976. Search in Google Scholar

[10] H. Bueno, L. P. Leme, and H. Rodrigues, Multiplicity of solutions for p-biharmonic problems with critical growth, Rocky Mountain J. Math. 48 (2018), 425–442, DOI: https://doi.org/10.1216/RMJ-2018-48-2-425. Search in Google Scholar

[11] N. T. Chung and P. H. Minh, Kirchhoff type problems involving p-biharmonic operators and critical exponents, J. Appl. Anal. Comput. 7 (2017), 659–669, DOI: https://doi.org/10.11918/2017041. Search in Google Scholar

[12] Z. Deng and Y. Huang, Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents, Appl. Math. Comput. 264 (2015), 323–334, DOI: https://doi.org/10.1016/j.amc.2015.04.099. Search in Google Scholar

[13] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Search in Google Scholar

[14] J. Chen and Z. Chen, Multiple Normalized Solutions for Biharmonic Choquard Equation with Hardy-Littlewood-Sobolev Upper Critical and Combined Nonlinearities, J. Geom. Anal. 33 (2023), 371, DOI: https://doi.org/10.1007/s12220-023-01428-1. Search in Google Scholar

[15] S. Liang, P. Pucci, and Z. Zhang, Multiple solutions for critical Choquard-Kirchhoff type equations, Adv. Nonlinear Anal. 10 (2020), 400–419, DOI: https://doi.org/10.1515/anona-2020-0119. Search in Google Scholar

[16] Y. Liu and M. Zhang, Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs, J. Math. Anal. Appl. 534 (2024), 128079, DOI: https://doi.org/10.1016/j.jmaa.2023.128079. Search in Google Scholar

[17] P. Pucci, M. Xiang, and B. Zhang, Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional p-Laplacian, Adv. Calc. Var 12 (2019), 253–275, DOI: https://doi.org/10.1515/acv-2016-0049. Search in Google Scholar

[18] Anu Rani and Sarika Goyal, Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions, Topol. Methods Nonlinear Anal. 59 (2022), 221–260, DOI: https://doi.org/10.12775/TMNA.2021.025. Search in Google Scholar

[19] H. Tao, L. Li, and P. Winkert, Existence and concentration of solutions for a 1-biharmonic Choquard equation with steep potential well in RN, J. Geom. Anal. 33 (2023), 276, DOI: https://doi.org/10.1007/s12220-023-01341-7. Search in Google Scholar

[20] S. Bai and D. D. Repovs, Existence and multiplicity Of solutions For critical Kirchhoff-Choquard equations involving the fractional p-Laplacian on the Heisenberg group, J. Nonlinear Var. Anal. 8 (2024), 143–166, DOI: https://doi.org/10.23952/jnva.8.2024.1.08. Search in Google Scholar

[21] W. Chen and D. Feng, Critical fractional (p, q)-Kirchhoff type problem with a nonline generalized Choquard nonlinearity and magnetic field, Bull. Malays. Math. Sci. Soc. 47 (2024), 33, DOI: https://doi.org/10.1007/s40840-023-01628-6. Search in Google Scholar

[22] S. Liu and H. Chen, Existence of ground-state solutions for p-Choquard equations with singular potential and doubly critical exponents, Math. Nachr. 296 (2023), 2467–2502, DOI: https://doi.org/10.1002/mana.202100255. Search in Google Scholar

[23] B. Yang, D. Zhang, and S. Liang, On critical double phase Choquard problems with singular nonlinearity, Commun. Nonlinear Sci. Numer. Simul. 125 (2023), 107420, DOI: https://doi.org/10.1016/j.cnsns.2023.107420. Search in Google Scholar

[24] Y. Zhang and D. Qin, Existence of solutions for a critical Choquard-Kirchhoff problem with variable exponents, J. Geom. Anal. 33 (2023), 200, DOI: https://doi.org/10.1007/s12220-023-01266-1. Search in Google Scholar

[25] Z. Shen, F. Gao, and M. Yang, Multiple solutions for nonhomogeneous Choquard equationinvolving Hardy-Littlewood-Sobolev critical exponent, Z. Angew. Math. Phys. 68 (2017), 61, DOI: https://doi.org/10.1007/s00033-017-0806-8. Search in Google Scholar

[26] P. Pucci, M. Xiang, and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN, Calc. Var. Partial Differential Equations 54 (2015), 2785–2806, DOI: https://doi.org/10.1007/s00526-015-0883-5. Search in Google Scholar

[27] X. Shi, Y. Zhao, and H. Chen, Existence of solutions for nonhomogeneous Choquard equations involving p-Laplacian, Mathematics 7 (2019), no. 9, 871, DOI: https://doi.org/10.3390/math7090871. Search in Google Scholar

[28] T. Wang and H. Guo, Infinitely many solutions for nonhomogeneous Choquard equations, Electron. J. Qual. Theory Differ. Equ. 24 (2019), 1–10, DOI: https://doi.org/10.14232/ejqtde.2019.1.24. Search in Google Scholar

[29] C. O. Alves, V. D. Râdulescu, and L. S. Tavares, Generalized Choquard equations driven by nonhomogeneous operators, Mediterr. J. Math. 16 (2019), 20, DOI: https://doi.org/10.1007/s00009-018-1287-5. Search in Google Scholar

[30] Y. Su and H. Chen, Existence of nontrivial solutions for a perturbation of Choquard equation with Hardy-Littlewood-Sobolev upper critical exponent, Electron. J. Differential Equations 2018 (2018), 1–25. Search in Google Scholar

[31] L. Wang, Multiple solutions for nonhomogeneous Choquard equations, Electron. J. Differential Equations 172 (2018), 1–27. Search in Google Scholar

[32] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl. 195 (2016), 2099–2129, DOI: https://doi.org/10.1007/s10231-016-0555-x. Search in Google Scholar

[33] M. Xiang, G. Molica Bisci, G. Tian, and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity 29 (2016), 357–374, DOI: https://doi.org/10.1088/0951-7715/29/2/357. Search in Google Scholar

[34] M. Xiang, B. Zhang, and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), 1021–1041, DOI: https://doi.org/10.1016/j.jmaa.2014.11.055. Search in Google Scholar

[35] A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. 69 (2008), 3629–3636, DOI: https://doi.org/10.1016/j.na.2007.10.001. Search in Google Scholar

[36] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315–334. Search in Google Scholar

[37] X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), 306–317, DOI: https://doi.org/10.1016/j.jmaa.2003.11.020. Search in Google Scholar

[38] V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184, DOI: https://doi.org/10.1016/j.jfa.2013.04.007Search in Google Scholar

[39] N. T. Chung and K. Ho, On a p(⋅)-biharmonic problem of Kirchhoff type involving critical growth, Appl. Anal. 101 (2021), 5700–5726, DOI: https://doi.org/10.1080/00036811.2021.1903445. Search in Google Scholar

[40] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case (I), Rev. Mat. Iberoam. 1 (1985), 145–201. Search in Google Scholar

[41] N. T. Chung, Multiple solutions for a non-cooperative elliptic system of Kirchhoff type involving p-biharmonic operator and critical growth, Acta Appl. Math. 165 (2020), 1–17, DOI: https://doi.org/10.1007/s10440-019-00237-5. Search in Google Scholar

[42] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490. Search in Google Scholar

[43] N. T. Chung, Existence of solutions for perturbed fourth order elliptic equations with variable exponents, Electron. J. Qual. Theory Differ. Equ. 96 (2018), 1–19, DOI: https://doi.org/10.14232/ejqtde.2018.1.96. Search in Google Scholar

[44] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. Search in Google Scholar

Received: 2024-06-23

Revised: 2024-11-21

Accepted: 2025-02-19

Published Online: 2025-03-25

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