Existence and nonexistence of solutions for elliptic problems with multiple critical exponents

Yuanyuan Li

doi:10.1515/math-2022-0605

Artikel Open Access

Existence and nonexistence of solutions for elliptic problems with multiple critical exponents

Yuanyuan Li

Veröffentlicht/Copyright: 29. Juli 2023

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Abstract

In this article, the existence and nonexistence of solutions for the quasilinear elliptic equations involving multiple critical terms under Dirichlet boundary conditions on bounded smooth domains Ω ⊂ R N ( N ≥ 3 ) are proved by using the variational method and Pohozaev identity, respectively.

Keywords: critical point; elliptic problem; existence; Pohozaev identity

MSC 2010: 35J20; 35D30

1 Introduction

This article is aimed to study the nontrivial weak solutions for the following elliptic problem:

(1) − Δ p u − ∑ i = 1 n μ i ∣ u ∣ p − 2 u ∣ x − a i ∣ p = ∣ u ∣ p ∗ ( s ) − 2 u ∣ x ∣ s + ∣ u ∣ p ∗ − 2 u + λ h ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω ,

where Ω ⊂ R N ( N ≥ 3 ) is a bounded smooth domain and the different points a i ∈ Ω , i = 1 , 2 , … , n . − Δ p u = − div ( ∣ ∇ u ∣ p − 2 ∇ u ) denotes the p -Laplacian of u with 1 < p < N . 0 ≤ μ 1 < μ 2 < … < μ n < μ ¯ ≔ N − p p p , and ∑ i = 1 n μ i < μ ¯ . λ is a positive parameter.

The corresponding energy functional for Problem (1) is defined by:

I ( u ) = 1 p ∫ Ω ∣ ∇ u ∣ p − ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a i ∣ p d x − 1 p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s d x − 1 p ∗ ∫ Ω ∣ u ∣ p ∗ d x − ∫ Ω λ H ( x , u ) d x ,

where H ( x , u ) = ∫ 0 u h ( x , t ) d t . The nontrivial weak solutions u ∈ W 0 1 , p ( Ω ) of Problem (1) are nonzero critical points of the functional I ( u ) . That is, u satisfies

(2) Φ ( u ) ≔ I ′ ( u ) u = ∫ Ω ∣ ∇ u ∣ p − ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a i ∣ p d x − ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s d x − ∫ Ω ∣ u ∣ p ∗ d x − ∫ Ω λ h ( x , u ) u d x = 0 .

In recent years, the elliptic problems have been studied. Brezis and Nirenberg [1] in 1983 considered elliptic problems with critical Sobolev terms and obtained positive solutions by using variational methods. Later, Struwe [2] showed the global compactness result on a bounded smooth domain. For the elliptic problems with multiple critical Hardy exponents, Li [3] considered Problem (1) without Sobolev-Hardy terms and obtained some existence results (for other related articles, please see [4,5]). For p -Laplace quasilinear elliptic equations with multiple critical terms (including critical Hardy terms and critical Sobolev-Hardy terms), there is little research up to now. Guo et al. in [6] and Chen in [7] studied the elliptic problem with a critical Sobolev-Hardy term and proved the existence and the multiplicity for nontrivial weak solutions by using variational methods. Other interesting results about elliptic equations can be seen in [8–12] and references therein.

By Ekeland’s variational principle (please see [13]) and some ideas from [3,14], we study the multiplicity for Problem (1) with multiple critical Hardy exponents and Sobolev-Hardy exponent. And it is much more difficulty in Problem (1) for the appearance of multiple critical exponents. In this article, we first consider three different nonempty sets that do not contain 0, and then under some assumptions, one critical point for the functional I ( u ) of Problem (1) in every set is proved. Hence, three different nontrivial weak solutions (positive, negative, and sign-changing solution) are obtained.

In the field of nonlinear elliptic problems, Pohozaev identity [15] plays a key role to study the existence and nonexistence of solutions. In 1965, Pohozaev [15] obtained the following results about the nonexistence of solution.

Pohozaev theorem. If Ω ⊂ R N is a star-shaped domain with respect to the origin, g ( u ) is a continuous function and satisfies λ N G ( u ) − N − 2 2 u g ( u ) ≤ 0 , where G ( u ) = ∫ 0 u g ( x ) d x , then there are no nontrivial solutions for the following problems:

− Δ u = λ g ( u ) , x ∈ Ω , u ∣ ∂ Ω = 0 .

In sprit of the Pohozaev theorem, we prove the nonexistence of solutions for Problem (1), please see Theorem 1.2 in this article.

In what follows, we denote

‖ u ‖ W ≔ ∫ Ω ∣ ∇ u ∣ p − ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a i ∣ p d x 1 p ,

where ∑ i = 1 n μ i < μ ¯ and u ∈ W 0 1 , p ( Ω ) , Then, by using the Hardy inequality (see [16]), we know that the norm ‖ u ‖ W is well defined and equivalent to the usual norm ∫ Ω ∣ ∇ u ∣ p d x 1 p in W 0 1 , p ( Ω ) .

Set

(3) A μ i , s = inf u ∈ W 0 1 , p ( Ω ) \ { 0 } ∫ Ω ∣ ∇ u ∣ p − μ i ∣ u ∣ p ∣ x − a i ∣ p ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ − s p p ∗ ( s ) ,

be the best Sobolev-Hardy constant, which can be achieved from [9,17].

In this article, we define

β + ≔ max { β , 0 } , β − ≔ max { − β , 0 } .

J 1 ≔ v ∈ W 0 1 , p ( Ω ) : ∫ Ω v + > 0 , Φ ( v + ) = 0 ; J 2 ≔ v ∈ W 0 1 , p ( Ω ) : ∫ Ω v − > 0 , Φ ( v − ) = 0 ; M 1 ≔ { v ∈ J 1 : v ≥ 0 } ; M 2 ≔ { v ∈ J 2 : v ≤ 0 } ; J 3 ≔ M 3 ≔ J 1 ⋂ J 2 .

The main theorems of this article are as follows.

Theorem 1.1

(Multiplicity of solutions) Assume

( h 0 ) : h ( x , u ) : Ω × R ↦ R is measurable in x and C 1 in u ;

( h 1 ) : h ( x , 0 ) = 0 , x ∈ Ω ;

( h 2 ) : a 3 ∥ u ∥ L d ( Ω ) d ≤ a 2 ∫ Ω H ( x , u ) ≤ ∫ Ω h ( x , u ) u ≤ a 1 ∫ Ω h u ( x , u ) u 2 ≤ a 4 ∥ u ∥ L d ( Ω ) d , where d ∈ ( p , p ∗ ) and u ∈ L d ( Ω ) . Constants a 1 ∈ 1 p ∗ − 1 , 1 p − 1 , a 2 ∈ ( p , p ∗ ) , 0 < a 3 < a 4 . Then, for any λ > λ ∗ ≔ λ ∗ ( N , p , d , a 3 ) , Problem (1) has three different nontrivial weak solutions (positive, negative, and sign-changing solution).

Theorem 1.2

(Nonexistence of solutions) Suppose Ω is a strictly star-shaped domain with respect to the origin, h ( x , u ) satisfies ( h 2 ) and

( h 3 ) : H ( x , u ) = 0 for x ∈ ∂ Ω ;

( h 4 ) : ∫ Ω ⟨ x , H x ⟩ ≤ 0 .

Moreover,

(4) ⟨ a i , x − a i ⟩ ≥ 0 , for x ∈ Ω ,

then Problem (1) has no nontrivial solutions.

2 Proof for existence of solutions

Recall that a sequence ( u m ) in spaces V is a Palais-Smale (PS) sequence for the functional I ( u ) if I ( u m ) → c , uniformly in m , and D I ( u m ) → 0 as m → ∞ . And if any PS sequence has a (strongly) convergent subsequence, then the functional I satisfies (PS) c condition.

First, from [18], we obtain that if

c < β ≔ min p − s p ( N − s ) A μ n , s N − s p − s , 1 N A 0 , 0 N p ,

the functional I ( u ) satisfies the ( PS ) c condition. Then, using the method in [3], ( PS ) c condition is also correct for the functional I ∣ M i ( i = 1 , 2 , 3 ) . That is, if u is a critical point of the restricted functional I ∣ M i , then we can deduce that u ∈ M i is also a critical point of the functional I ( u ) for Problem (1). Thus, u is a weak solution for (1).

Now, we prove that I ( u ) has a mountain pass geometry.

Lemma 2.1

For any u 0 ∈ W 0 1 , p ( Ω ) , u 0 > 0 ( u 0 < 0 ) , there exists τ λ > 0 such that τ λ u 0 ∈ J 1 ( ∈ J 2 ) . Furthermore, lim λ → ∞ τ λ = 0 .

Thus, there exists τ 1 λ , τ 2 λ > 0 such that τ 1 λ u 0 + τ 2 λ u 1 ∈ J 3 , for u 0 , u 1 ∈ W 0 1 , p ( Ω ) , u 0 > 0 and u 1 < 0 with disjoint supports. Moreover, τ 1 λ , τ 2 λ → 0 as λ → ∞ .

Proof

From Definition (2) of Φ ( u ) , we only need to prove that Φ ( τ λ u 0 ) = 0 for u 0 > 0 and some τ λ > 0 . In fact, by the condition ( h 2 ) , we deduce

Φ ( τ u 0 ) ≥ A 1 τ p − A 2 t p ∗ − A 3 τ p ∗ ( s ) − λ a 4 A 4 τ d ,

where A 1 = ‖ u ‖ W p , A 2 = ∫ Ω ∣ u 0 ∣ p ∗ , A 3 = ∫ Ω ∣ u 0 ∣ p ∗ ( s ) ∣ x ∣ s , A 4 = ∫ Ω ∣ u 0 ∣ d . On the other hand,

Φ ( τ u 0 ) ≤ A 1 τ p − A 2 τ p ∗ − A 3 τ p ∗ ( s ) − λ a 3 A 4 τ d .

From Bolzano’s theorem and p < d < p ∗ , we know that for some τ = τ λ ,

Φ ( τ λ u ) = 0 .

Then, from Φ ( τ u 0 ) ≤ A 1 τ p − λ a 3 A 4 τ d , there exists τ 1 = A 1 a 3 A 4 λ 1 d − p such that A 1 τ 1 p − λ a 3 A 4 τ 1 d = 0 . And hence, we finish the proof as τ λ ∈ [ 0 , τ 1 ] .□

Lemma 2.2

For any u ∈ M i ( i = 1 , 2 , 3 ) , there exist B 1 , B 2 > 0 , such that

B 1 ‖ u ‖ W p ≤ I ( u ) ≤ B 2 ‖ u ‖ W p .

Proof

From u ∈ M i , we can easily calculate that

‖ u ‖ W p − ∫ Ω ∣ u ∣ p ∗ = ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s + λ ∫ Ω h ( x , u ) u .

On the one hand, from ( h 2 ) and I ′ ( u ) u = 0

I ( u ) = 1 p ‖ u ‖ W p − 1 p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s − 1 p ∗ ∫ Ω ∣ u ∣ p ∗ − ∫ Ω λ H ( x , u ) = 1 p − 1 p ∗ ∫ Ω ∣ u ∣ p ∗ + 1 p − 1 p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s + λ p ∫ Ω h ( x , u ) u − ∫ Ω λ H ( x , u ) ≥ 1 p − 1 p ∗ ∫ Ω ∣ u ∣ p ∗ + 1 p − 1 p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s + 1 p − 1 a 2 λ ∫ Ω h ( x , u ) u .

Thus, from p < a 2 < p ∗ , the first inequality is proved.

On the other hand, using ( h 1 ) and ( h 2 ) , we can prove that

I ( u ) = 1 p ‖ u ‖ W p − 1 p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s − 1 p ∗ ∫ Ω ∣ u ∣ p ∗ − ∫ Ω λ H ( x , u ) ≤ 1 p ‖ u ‖ W p .

And hence, we finish the proof.□

Lemma 2.3

‖ u ± ‖ W ≥ C , for any u ∈ M i ( i = 1 , 2 , 3 ) , and

I ( u ) ≥ C ‖ u ‖ W p , ∀ u ∈ W 1 , p ( Ω ) ,

if ‖ u ‖ W small enough for some constant C > 0 .

Proof

For u ∈ M i ( i = 1 , 2 , 3 ) , from ( h 1 ) and ( h 2 ), we calculate that

‖ u ± ‖ W p = I ( u ± ) ≤ a 1 ‖ u ± ‖ L d ( Ω ) l + ‖ u ± ‖ L p ∗ ( Ω ) p ∗ + a 2 ‖ u ± ‖ W p ∗ ( s ) ≤ a 3 ‖ u ± ‖ W d + a 4 ‖ u ± ‖ W p ∗ + a 2 ‖ u ± ‖ W p ∗ ( s ) .

Thus, ‖ u ± ‖ ≥ C from p < d < p ∗ and p < p ∗ ( s ) .

On the other hand, with ( h 2 )

I ( u ) = 1 p ‖ u ‖ W p − 1 p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s − 1 p ∗ ∫ Ω ∣ u ∣ p ∗ − ∫ Ω λ H ( x , u ) ≥ 1 p ‖ u ‖ W p − 1 p ∗ ‖ u ‖ L p ∗ p ∗ − D ‖ u ‖ L d d − D 5 ‖ u ‖ W p ∗ ( s ) ≥ 1 p ‖ u ‖ W p − 1 p ∗ c 1 ‖ u ‖ W p ∗ − c 2 ‖ u ‖ W d − D 5 ‖ u ‖ W p ∗ ( s ) = ‖ u ‖ W p 1 p − 1 p ∗ c 1 ‖ u ‖ W p ∗ − p − c 2 ‖ u ‖ W d − p − D 5 ‖ u ‖ W p ∗ ( s ) − p .

Combining p < d < p ∗ and p < p ∗ ( s ) , we can prove that I ( u ) ≥ C ‖ u ‖ W p for ‖ u ‖ W small enough.□

Lemma 2.4

The set J 1 and J 2 belong to C 1 sub-manifold of W 1 , p ( Ω ) with co-dimension 1 and J 3 is a C 1 sub-manifold of W 1 , p ( Ω ) with co-dimension 2. Furthermore, the sets M i ( i = 1 , 2 , 3 ) are complete. Moreover,

T u W 0 1 , p ( Ω ) = T u J 1 ⊕ span { u + } , T u W 0 1 , p ( Ω ) = T u J 2 ⊕ span { u − } , T u W 0 1 , p ( Ω ) = T u J 3 ⊕ span { u + , u − } ,

for any u ∈ J i , where T w B denotes the tangent space at w for the manifold B. And it is uniformly continuous on bounded sets of J i ( i = 1 , 2 , 3 ) for the projection onto T u J i in this decomposition.

Proof

Now, we prove that J i belongs to a C 1 sub-manifold. Set

J ¯ 1 = u ∈ W 0 1 , p ( Ω ) : ∫ Ω v + > 0 ; J ¯ 2 = u ∈ W 0 1 , p ( Ω ) : ∫ Ω v − > 0 ; J ¯ 3 = J ¯ 1 ⋂ J ¯ 2 .

From the definitions, we know that J i is a subset of J ¯ i ( i = 1 , 2 , 3 ) , which are open in the space W 1 , p ( Ω ) .

As Lemma 5 in [14], we assume f i : J ¯ i ⟶ R k , with k = 1 ( i = 1 , 2 ) , k = 2 ( i = 3 ) is a C 1 function. Then, the inverse image of a regular value for f i is J i . Indeed, for u ∈ J ¯ 1 ,

f 1 ( u ) = I ′ ( u + ) u + ;

for u ∈ J ¯ 2 ,

f 2 ( u ) = I ′ ( u − ) u − ;

for u ∈ J ¯ 3 ,

f 3 ( u ) = ⟨ f 1 ( u ) , f 2 ( u ) ⟩ ,

where ⟨ . , . ⟩ denotes the duality pairing of the space W 1 , p ( Ω ) . Thus, J i = f i − 1 ( 0 ) . Following [19], we deduce that f i is a C 1 function. Now, we need to prove that a regular value of f i is 0. Combining p < p ∗ ( s ) < p ∗ and 1 p ∗ − 1 < a 1 < 1 p − 1 , for u ∈ J 1 and from ( h 2 ), we deduce that

⟨ f 1 ′ ( u + ) , u + ⟩ = p ‖ u + ‖ W p − p ∗ ‖ u + ‖ L p ∗ ( Ω ) p ∗ − p ∗ ( s ) ∫ Ω ∣ u + ∣ p ∗ ( s ) ∣ x ∣ s − λ ∫ Ω ( h ( x , u ) u + + h u ( x , u ) u + 2 ) = p I ( u + ) − p ∗ ‖ u + ‖ L p ∗ p ∗ − p ∗ ( s ) ∫ Ω ∣ u + ∣ p ∗ ( s ) ∣ x ∣ s − λ ∫ Ω ( h ( x , u ) u + + h u ( x , u ) u + 2 ) = ( p − p ∗ ) ‖ u + ‖ L p ∗ p ∗ + ( p − p ∗ ( s ) ) ∫ Ω ∣ u + ∣ p ∗ ( s ) ∣ x ∣ s + p ∫ Ω λ h ( x , u ) u + − λ ∫ Ω ( h ( x , u ) u + + h u ( x , u ) u + 2 ) ≤ λ ( p − 1 ) ∫ Ω h ( x , u ) u + − λ ∫ Ω h u ( x , u ) u + 2 ≤ λ a 1 p − 1 − 1 a 1 ∫ Ω h u ( x , u ) u + 2 ≤ λ a 1 a 4 p − 1 − 1 a 1 ‖ u + ‖ L l l < 0 .

Hence, J 1 belongs to a C 1 sub-manifold. Similarly, it is correct for the case of J 2 . As for the case of u ∈ J 3 , we obtain

⟨ f 1 ′ ( u ) , u − ⟩ = ⟨ f 2 ′ ( u ) , u + ⟩ = 0 .

The case for J 3 is also correct.

Similarly, by using Lemma 2.2, as Lemma 5 in [14], the remainder can be proved and we omit it here.□

Next, we can prove Theorem 1.1.

Proof of Theorem 1.1

For u ∈ M i , we know that the restricted functional I ( u ) ∣ M i is bounded from below. Thus, there exists a sequence w k ∈ M i from Ekeland’s variational principle such that

I ( w k ) → β i ≔ inf M i I ( u ) , ( I ∣ M i ) ′ ( w k ) → 0 .

From Lemma 2.1, there exists w 0 > 0 such that

β 1 ≤ I ( τ λ w 0 ) ≤ 1 p τ λ p ‖ u ‖ W p .

With Lemma 2.3, we deduce β 1 → 0 as λ → ∞ . Then, using Lemma 2.1 again, we can choose λ > λ ∗ ( N , p , l , a 3 ) satisfying

β 1 < β .

We obtain that { w k } has a convergent subsequence { w k j } . Hence, there is a critical point for I ( u ) in the manifold M i ( i = 1 , 2 , 3 ) . Then, we finish the proof.□

3 Proof for nonexistence of solutions

We prove the nonexistence of solutions by using the Pohozaev identity and following some ideas from [20–22].

Proof of Theorem 1.2

Multiplying equation (1) by ⟨ x , ∇ u ⟩ , we calculate that

(5) ∫ Ω − Δ p u − ∑ i = 1 n μ i ∣ u ∣ p − 2 u ∣ x − a i ∣ p ⟨ x , ∇ u ⟩ = ∫ Ω ∣ u ∣ p ∗ ( s ) − 2 u ∣ x ∣ s + ∣ u ∣ p ∗ − 2 u + λ h ( x , u ) ⟨ x , ∇ u ⟩ .

After carefully calculating, we obtain

(6) ∫ Ω Δ p u ⟨ x , ∇ u ⟩ = p − 1 p ∫ ∂ Ω ∣ ∇ u ∣ p ⟨ x , ν ⟩ d S − p − N p ∫ Ω ∣ ∇ u ∣ p .

(7) ∫ Ω μ i ∣ u ∣ p − 2 u ∣ x − a i ∣ p ⟨ x , ∇ u ⟩ = − ∫ Ω N p μ i ∣ u ∣ p ∣ x − a ∣ p + ∫ Ω μ i ∣ u ∣ p ∣ x − a ∣ p + 2 x i ( x i − a i ) = − N − p p ∫ Ω μ i ∣ u ∣ p ∣ x − a ∣ p + ∫ Ω μ i ∣ u ∣ p ∣ x − a ∣ p + 2 ⟨ a , x − a ⟩ .

Again,

(8) ∫ Ω ∣ u ∣ p ∗ ( s ) − 2 u ∣ x ∣ s ⟨ x , ∇ u ⟩ = ∫ Ω s − N p ∗ ( s ) ∣ u ∣ p ∗ ( s ) ∣ x ∣ s .

(9) ∫ Ω ∣ u ∣ p ∗ − 1 ⟨ x , ∇ u ⟩ = − N p ∗ ∫ Ω ∣ u ∣ p ∗ .

And with ( h 3 )

(10) ∫ Ω λ h ( x , u ) ⟨ x , ∇ u ⟩ = ∫ ∂ Ω λ H ( x , u ) ⟨ x , v ⟩ d S − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ = − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ .

Then for (2), we have

(11) ∫ Ω ∣ ∇ u ∣ p − ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a i ∣ p = ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s + ∫ Ω ∣ u ∣ p ∗ + ∫ Ω λ h ( x , u u ) .

Thus, replacing (5) with (6)–(10), we obtain that

− p − 1 p ∫ ∂ Ω ∣ ∇ u ∣ p ⟨ x , ν ⟩ d S + p − N p ∫ Ω ∣ ∇ u ∣ p + N − p p ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p − ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p + 2 ⟨ a , x − a ⟩ = ∫ Ω s − N p ∗ ( s ) ∣ u ∣ p ∗ ( s ) ∣ x ∣ s − N p ∗ ∫ Ω ∣ u ∣ p ∗ − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ .

Therefore, from (4), (11) and ( h 4 ) ,

0 > − p − 1 p ∫ ∂ Ω ∣ ∇ u ∣ p ⟨ x , ν ⟩ d S = N − p p ∫ Ω ∣ ∇ u ∣ p − N − p p ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p + ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p + 2 ⟨ a , x − a ⟩ + ∫ Ω s − N p ∗ ( s ) ∣ u ∣ p ∗ ( s ) ∣ x ∣ s − N p ∗ ∫ Ω ∣ u ∣ p ∗ − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ = N − p p ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s + N − p p ∫ Ω ∣ u ∣ p ∗ + N − p p ∫ Ω λ h ( x , u ) u + ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p + 2 ⟨ a , x − a ⟩ + ∫ Ω s − N p ∗ ( s ) ∣ u ∣ p ∗ ( s ) ∣ x ∣ s − N p ∗ ∫ Ω ∣ u ∣ p ∗ − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ = N − p p + s − N p ∗ ( s ) ∫ Ω ∣ u ∣ p ∗ ( s ) ∣ x ∣ s + N − p p − N p ∗ ∫ Ω ∣ u ∣ p ∗ + N − p p ∫ Ω λ h ( x , u ) u + ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p + 2 ⟨ a , x − a ⟩ − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ = ∫ Ω ∑ i = 1 n μ i ∣ u ∣ p ∣ x − a ∣ p + 2 ⟨ a , x − a ⟩ + N − p p ∫ Ω λ h ( x , u ) u − N ∫ Ω λ H ( x , u ) − ∫ Ω λ ⟨ x , ∇ H x ⟩ ≥ 0

It is a contradiction, and hence, the proof is finished.□

Acknowledgement

The authors are sincerely grateful to the anonymous referees for their careful comments.

Funding information: This article was supported by National Natural Science Foundation of China (No. 12101236).
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-03-25

Revised: 2023-06-22

Accepted: 2023-06-23

Published Online: 2023-07-29

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https://doi.org/10.1515/math-2022-0605

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critical point; elliptic problem; existence; Pohozaev identity

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