Bifurcation and multiplicity results for critical problems involving the p-Grushin operator

Paolo Malanchini; Giovanni Molica Bisci; Simone Secchi

doi:10.1515/anona-2025-0089

Article Open Access

Bifurcation and multiplicity results for critical problems involving the p-Grushin operator

Paolo Malanchini , Giovanni Molica Bisci and Simone Secchi

Published/Copyright: August 8, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we prove a bifurcation and multiplicity result for a critical problem involving a degenerate nonlinear operator Δ γ p . We extend to a generic p > 1 a result, which was proved only when p = 2 . When p ≠ 2 , the nonlinear operator − Δ γ p has no linear eigenspaces, so our extension is nontrivial and requires an abstract critical theorem, which is not based on linear subspaces. We use an abstract result based on a pseudo-index related to the Z 2 -cohomological index that is applicable here. We provide a version of the Lions’ concentration-compactness principle for our operator.

Keywords: p-Grushin operator; critical exponent

MSC 2010: 35J20; 35J70

1 Introduction

Suppose that N ≥ 3 , and that N = m + ℓ for some positive integers m and ℓ . Let z = ( x , y ) be a generic point of R N = R m × R ℓ , and let γ ≥ 0 be a real parameter. The Grushin-Baouendi operator is defined as follows:

(1) Δ γ u ( z ) = Δ x u ( z ) + ∣ x ∣ 2 γ Δ y u ( z ) ,

where Δ x and Δ y are the Laplace operators in the variables x and y , respectively.

It follows from (1) that the Grushin operator is not uniformly elliptic in the space R N since it is degenerate on the subspace { 0 } × R ℓ . For more technical reasons, see the survey [8], the Grushin operator belongs to the class of subelliptic operators, which lies halfway between elliptic and hyperbolic operators.

As remarked in [15] (see also [1]), if γ is a nonnegative integer, the operator Δ γ falls into the class of Hörmander operators, which are defined to be a sum of squares of vector fields generating a Lie algebra of maximum rank at any point of R N . If we introduce the family of vector fields

X i = ∂ ∂ x i , i = 1 , … , n , X m + j = ∣ x ∣ γ ∂ ∂ y j , j = 1 , … , ℓ ,

the corresponding Grushin gradient is

∇ γ = ( ∇ x , ∣ x ∣ γ ∇ y ) = ∂ ∂ x i , … , ∂ ∂ x m , ∣ x ∣ γ ∂ ∂ y 1 , … , ∣ x ∣ γ ∂ ∂ y ℓ ,

and the Grushin operator can be written as follows:

Δ γ = ∑ i = 1 N X i 2 = ∇ γ ⋅ ∇ γ .

In a similar manner to how the Grushin operator can be constructed from the Laplacian, it is possible to define a quasilinear Grushin-type operator based on the p -Laplace operator. The p -Laplace Grushin operator ( p -Grushin for short) is defined by

(2) Δ γ p u = ∇ γ ⋅ ( ∣ ∇ γ u ∣ p − 2 ∇ γ u ) .

To the best of our knowledge, this operator has been introduced for the first time by Dou and Han in [7]. The p -Grushin operator has obtained increasing attention in recent years. Recent works, such as those by Huang and Yang [13] and Huang et al. [14], have explored the existence, uniqueness, and regularity of solutions to the p -Laplace equation involving Grushin-type operators. Moreover, in [27], the authors proved a Liouville-type theorem for stable solutions of weighted p -Laplace-type Grushin equations.

In this article, we are concerned with a critical problem associated to the operator (2). More precisely, we consider the boundary value problem:

(3) − Δ γ p u = λ ∣ u ∣ p − 2 u + ∣ u ∣ p γ * − 2 u in Ω , u = 0 on ∂ Ω ,

where Ω ⊂ R N is bounded with 0 ∈ Ω ¯ and p γ * is the critical Sobolev exponent for the p -Grushin operator defined in (6) below.

In the seminal paper [5], Brézis and Nirenberg considered the problem with the Sobolev critical growth

(4) − Δ u = λ u + ∣ u ∣ 2 * − 2 u in Ω , u = 0 on ∂ Ω ,

on a bounded domain Ω ⊂ R N . Here, N ≥ 3 , 2 * = 2 N ⁄ ( N − 2 ) is the critical Sobolev exponent, and λ > 0 is a positive parameter. By a careful comparison argument, the authors proved that for N ≥ 4 ,

inf u ∈ H 0 1 ( Ω ) ∫ Ω ∣ u ∣ 2 * = 1 ∫ Ω ∣ ∇ u ∣ 2 − λ ∫ Ω ∣ u ∣ 2 < inf u ∈ H 0 1 ( Ω ) ∫ Ω ∣ u ∣ 2 * = 1 ∫ Ω ∣ ∇ u ∣ 2 = S ,

where S is the best constant for the continuous Sobolev embedding H 0 1 ( Ω ) ⊂ L 2 * ( Ω ) . Calling λ 1 the first eigenvalue of the operator − Δ defined on H 0 1 ( Ω ) ⊂ L 2 ( Ω ) , the existence of a positive solution to (4) follows for every λ ∈ ( 0 , λ 1 ) if N ≥ 4 . The case N = 3 is harder, and the authors proved the existence of a positive solution when Ω is a ball and λ is sufficiently close to λ 1 .

The article [5] marked the beginning of an endless stream of efforts to extend the seminal results. In 1984, Cerami et al. proved in [6, Theorem 1.1] the following multiplicity result for problem (4). Let 0 < λ 1 < λ 2 ≤ λ 3 ≤ ⋯ → + ∞ be the Dirichlet eigenvalues of − Δ in Ω . If λ k ≤ λ < λ k + 1 and

λ > λ k + 1 − S ∣ Ω ∣ 2 ⁄ N ,

then (4) has m distinct pairs of nontrivial solutions ± u j λ , where m is the multiplicity of the eigenvalue λ k + 1 .

Similar multiplicity and bifurcation results have been proved for several classes of variational operators. For instance, the authors considered in [10] a rather general family of fractional (i.e., nonlocal) operators, which contains the usual fractional Laplace operator in R N .

Subsequently, a multiplicity result has been proved in [23] for the corresponding problem for the p -Laplacian

− Δ p u = λ ∣ u ∣ p − 2 u + ∣ u ∣ p * − 2 u in Ω , u = 0 on ∂ Ω ,

where Δ p u = ∇ ⋅ ( ∣ ∇ u ∣ p − 2 ∇ u ) is the p -Laplace operator and p * = N p ⁄ ( N − p ) .

Finally, the authors in [22] extended the same multiplicity result for a critical problem involving the fractional p -Laplace operator.

The case p = 2 of (3) was studied in [2], where the authors extended the Brezis-Nirenberg result to the critical problem with the Grushin operator. A multiplicity result for the case p = 2 was proved in [3].

In this article, we extend the above bifurcation and multiplicity results to the problem (3). This extension requires some care. Indeed, the linking argument based on the eigenspace of − Δ γ in [3] does not work when p ≠ 2 since the nonlinear operator − Δ γ p does not have linear eigenspaces. We will use a different sequence of eigenvalues that is based on the Z 2 -cohomological index of Fadell and Rabinowitz [9].

In what follows, S stands for the optimal constant for the Sobolev embedding to be defined in (7), and ∣ Ω ∣ denotes the Lebesgue measure of a bounded domain Ω ⊂ R N with 0 ∈ Ω ¯ . If we denote by { λ k } the sequence of eigenvalues of the p -Grushin operator introduced in Section 4, we can state the main result of this article.

Theorem 1.1

The following facts hold:

If
λ 1 − S ∣ Ω ∣ p ⁄ N γ < λ < λ 1 ,
then problem (3) has a pair of nontrivial solutions ± u λ such that u λ → 0 as λ ↗ λ 1 .
If λ k ≤ λ < λ k + 1 = … = λ k + m < λ k + m + 1 for some k , m ∈ N and
(5) λ > λ k + 1 − S ∣ Ω ∣ p ⁄ N γ ,
then problem (3) has m distinct pairs of nontrivial solutions ± u j λ , j = 1 , … , m , such that u j λ → 0 as λ ↗ λ k + 1 .

In particular, we have the following existence result.

Corollary 1.2

Problem (3) has a nontrivial solution for all

λ ∈ ⋃ k = 1 ∞ ( λ k − S ⁄ ∣ Ω ∣ p ⁄ N γ , λ k ) .

The article is organized as follows. In Section 2, we introduce the functional space W ˚ γ 1 , p ( Ω ) . In Section 3, we prove a concentration-compactness principle for the p -Grushin operator. Section 4 focuses on the eigenvalues of the operator − Δ p . Section 5 recalls an abstract multiplicity result, which plays a key role in the proof of Theorem 1.1, presented in Section 6.

2 Functional setting

Fix a bounded domain Ω ⊆ R N and a number 1 < p < ∞ . The Sobolev space W ˚ γ 1 , p ( Ω ) is defined as the completion of C c 1 ( Ω ) with respect to the norm

∥ u ∥ γ , p = ∫ Ω ∣ ∇ γ u ∣ p d z 1 ⁄ p .

Specially, the space W ˚ γ 1,2 ( Ω ) = H ( Ω ) is a Hilbert space endowed with the inner product

⟨ u , v ⟩ γ = ∫ Ω ∇ γ u ∇ γ v d z .

The following embedding result was proved in [15, Proposition 3.2 and Theorem 3.3]. See also [13, Corollary 2.11].

Proposition 2.1

Let Ω ⊂ R N be a bounded open set. Then the embedding

W ˚ γ 1 , p ( Ω ) ↪ L q ( Ω )

is compact for every q ∈ [ 1 , p γ * ) , where

(6) p γ * = p N γ N γ − p

and

N γ = m + ( 1 + γ ) ℓ

is the homogeneous dimension of R N associated with the decomposition R N = R m × R ℓ .

We may thus define the best constant of the Sobolev embedding W ˚ γ 1 , p ( Ω ) ↪ L p γ * ( Ω )

(7) S = inf u ∈ W ˚ γ 1 , p ( Ω ) u ≠ 0 ∫ Ω ∣ ∇ γ u ∣ p d z ∫ Ω ∣ u ∣ p γ * d z p ⁄ p γ * .

3 Concentration-compactness

In this section, we adapt the arguments from [18,19] and [26, Theorem 4.8] to establish a concentration-compactness result for the p -Grushin operator.

Let ℳ ( R N ) be the space of all finite signed Radon measures. We recall the definition of tight convergence of measures.

Definition 3.1

A sequence of measures { μ n } ⊂ ℳ ( R N ) converges tightly to a measure μ ∈ ℳ ( R N ) , that is, μ n ⇀ ∗ μ , if for every φ ∈ C b ( R N )

∫ R N φ d μ n → ∫ R N φ d μ as n → ∞ ,

where C b ( R N ) is the space of the bounded, continuous functions on R N .

We will also need the following lemma, see [18, Lemma 1.2 and Remark 1.5].

Lemma 3.2

Let μ , ν two bounded nonnegative measures on Ω satisfying for some constant C 0 ≥ 0

∫ Ω ∣ φ ∣ q d ν 1 ⁄ q ≤ C 0 ∫ Ω ∣ φ ∣ p d μ 1 ⁄ p ∀ φ ∈ C c ∞ ( Ω ) ,

where 1 ≤ p ≤ q ≤ + ∞ . Then, there exist an at most countable set J, families { z j } j ∈ J of distinct points in Ω ¯ and positive numbers { ν j } j ∈ J such that

ν = ∑ j ∈ J ν j δ z j , μ ≥ C 0 − p ∑ j ∈ J ν j p ⁄ q δ z j .

Now we are ready to prove the concentration-compactness principle.

Theorem 3.3

Let Ω be a bounded subset in R N and { u n } be a bounded sequence in W ˚ γ 1 , p ( Ω ) . Then we have:

up to a subsequence, there exist u ∈ W ˚ γ 1 , p ( Ω ) , two bounded nonnegative measures μ and ν , an at most countable set J, a family { z j } j ∈ J of distinct points in Ω ¯ and a family { ν j } j ∈ J of positive numbers such that
(8) u n ⇀ u in W ˚ γ 1 , p ( Ω ) , μ n ≔ ∣ ∇ γ u n ∣ p ⇀ ∗ μ , ν n ≔ ∣ u n ∣ p γ * ⇀ ∗ ν ,

(9) ν = ∣ u ∣ p γ * + ∑ j ∈ J ν j δ z j ,
where δ z is the Dirac measure concentrated at z ∈ R N .
In addition, we have
(10) μ ≥ ∣ ∇ γ u ∣ p + ∑ j ∈ J μ j δ z j
for some family of positive numbers { μ j } j ∈ J satisfying
(11) μ j ≥ S ν j p p γ * f o r a l l j ∈ J ,
where S is defined in (7). In particular,
∑ j ∈ J ( ν j ) p p γ * < ∞ .

Proof

Since Ω is bounded, the measures ∣ ∇ γ u n ∣ p and ∣ u n ∣ p γ * are uniformly tight in n . Therefore, by Prohorov’s theorem [11, Theorem 1.208], there exist two nonnegative bounded measures μ , ν in Ω such that (8) holds.

Let φ ∈ C c ∞ ( Ω ) , by Sobolev inequalities, we have

∫ Ω ∣ φ u n ∣ p γ * d z ≤ S − p γ * ⁄ p ∫ Ω ∣ ∇ γ ( φ u n ) ∣ p d z 1 ⁄ p .

As n → ∞ , the left-hand side converges to ∫ Ω ∣ φ ∣ p γ * d ν , while the right-hand side approaches S − p γ * ⁄ p ∫ Ω ∣ φ ∣ p d μ 1 ⁄ p , since all the remaining lower-order terms in the expansion of ∣ ∇ γ ( φ u n ) ∣ p converge to zero in L p ( Ω ) as n → ∞ because of the compactness of the Sobolev embeddings.

That is, there holds

(12) ∫ Ω ∣ φ ∣ p γ * d ν 1 ⁄ p γ * ≤ S − 1 ⁄ p ∫ Ω ∣ φ ∣ p d μ 1 ⁄ p

for all φ ∈ C c ∞ ( Ω ) . So Theorem 3.3 is proved in the case u ≡ 0 applying Lemma 3.2.

We now turn to the general case u n ⇀ u . Let v n = u n − u ∈ W ˚ γ 1 , p ( Ω ) . Then v n ⇀ 0 in W ˚ γ 1 , p ( Ω ) , and by the Brezis-Lieb Lemma (see [4, Theorem 1]),

(13) ∫ Ω ∣ u n ∣ p γ * d z − ∫ Ω ∣ u n − u ∣ p γ * d z → ∫ Ω ∣ u ∣ p γ * d z

as n → ∞ . Moreover, by (13), we have

ω n ≔ ν n − ∣ u ∣ p γ * = ∣ u n ∣ p γ * − ∣ u ∣ p γ * = ∣ u n − u ∣ p γ * + o ( 1 ) = ∣ v n ∣ p γ * + o ( 1 ) .

Also let λ n = ∣ ∇ γ v n ∣ 2 . We may assume that λ n ⇀ ∗ λ , while ω n ⇀ ∗ ω = ν − ∣ u ∣ p γ * , where λ , ω are positive measures.

Consider φ ∈ C c ∞ ( Ω ) . Then, arguing as earlier,

∫ Ω ∣ φ ∣ p γ * d ω = lim n → ∞ ∫ Ω ∣ φ ∣ p γ * d ω n = lim n → ∞ ∫ Ω ∣ v n φ ∣ p γ * d z ≤ S − p γ * ⁄ p liminf n → ∞ ∫ Ω ∣ ∇ γ ( v n φ ) ∣ p d z p γ * ⁄ p = S − p γ * ⁄ p liminf n → ∞ ∫ Ω ∣ φ ∣ p ∣ ∇ γ v n ∣ p d z p γ * ⁄ p = S − p γ * ⁄ p ∫ Ω ∣ φ ∣ p d λ p γ * ⁄ p .

That is, there holds

∫ Ω ∣ φ ∣ p γ * d ω 1 ⁄ p γ * ≤ S − 1 ⁄ p ∫ Ω ∣ φ ∣ p d λ 1 ⁄ p

for all φ ∈ C c ∞ ( Ω ) . Now (9) holds true by Lemma 3.2.

To obtain (10), we first claim that μ ≥ ∣ ∇ γ u ∣ p . Indeed, for any φ ∈ C c ∞ ( Ω ) , φ ≥ 0 , the functional

v ↦ ∫ Ω ∣ ∇ γ v ∣ p φ d z

is convex and continuous, therefore, u n ⇀ u in W ˚ γ 1 , p ( Ω ) implies

∫ Ω φ d μ = lim n → ∞ ∫ Ω ∣ ∇ γ u n ∣ p φ d z ≥ ∫ Ω ∣ ∇ γ u ∣ p φ d z , for any φ ∈ C c ∞ ( Ω ) , φ ≥ 0 .

Let φ ∈ C c ∞ ( Ω ) such that 0 ≤ φ ≤ 1 , supp φ = B ( 0 , 1 ) and φ ( 0 ) = 1 . Given ε > 0 , we apply the inequality (12) with φ z − z j ε , where j is fixed in J . We obtain

ν j 1 ⁄ p γ * S 1 ⁄ p ≤ μ ( B ( z j , ε ) ) 1 ⁄ p .

This implies that μ ( { z j } ) > 0 and

μ ≥ ν j p ⁄ p γ * S δ z j ,

and thus,

μ ≥ ∑ j ∈ J ν j p ⁄ p γ * S δ z j .

Since { ∣ ∇ γ u ∣ p } ∪ { δ z j ∣ j ∈ J } is a set consisting of pairwise mutually singular measures and the latter estimate holds, (10) and (11) follow.□

4 Eigenvalues of the p -Grushin operator

The Dirichlet spectrum of − Δ γ p in Ω consists of those λ ∈ R for which the problem

(14) − Δ γ p u = λ ∣ u ∣ p − 2 u in Ω , u = 0 on ∂ Ω ,

has a nontrivial weak solution u ∈ W ˚ γ 1 , p ( Ω ) .

The spectrum of the 2-Grushin operator was studied in [28, Theorem 1]. It consists of a sequence { λ k } such that

0 < λ 1 < λ 2 ≤ … ≤ λ k ≤ λ k + 1 ≤ …

and

λ k → + ∞ as k → + ∞ .

A complete description of the spectrum remains unknown even for the classical p -Laplacian case. It is well known that the first eigenvalue λ 1 is positive, simple, and has an associate positive eigenfunction φ 1 , see [16,17]. Increasing and unbounded sequences of eigenvalues can be defined using various minimax schemes, but a complete list of the eigenvalues of − Δ p remains unavailable.

Let us denote by W γ − 1 , p ′ ( Ω ) = ( W ˚ γ 1 , p ( Ω ) ) * the dual space of W ˚ γ 1 , p ( Ω ) . Introducing the operator A p : W ˚ γ 1 , p ( Ω ) → W γ − 1 , p ′ ( Ω ) by

⟨ A p ( u ) , v ⟩ = ∫ Ω ∣ ∇ γ u ∣ p − 2 ∇ γ u ∇ γ v d z ,

a weak solution of (14) can be characterized as a function u ∈ W ˚ γ 1 , p ( Ω ) such that

⟨ A p ( u ) , v ⟩ = λ ∫ Ω ∣ u ∣ p − 2 u v d z

for all v ∈ W ˚ γ 1 , p ( Ω ) .

We collect here some remarkable properties of the nonlinear operator A p ∈ C ( W ˚ γ 1 , p ( Ω ) , W γ − 1 , p ′ ( Ω ) ) .

A p is ( p − 1 ) -homogeneous^[1] and odd;
A p is uniformly positive: ⟨ A p ( u ) , u ⟩ = ∥ u ∥ γ , p p for each u ∈ W ˚ γ 1 , p ( Ω ) ;
A p is a potential operator: in fact, it is the Fréchet derivative of the functional u ↦ ∥ u ∥ γ , p p p in W ˚ γ 1 , p ( Ω ) ;
A p is of type ( S ) : every sequence { u j } in W ˚ γ 1 , p ( Ω ) such that
u j ⇀ u , ⟨ A p ( u j ) , u j − u ⟩ → 0
has a subsequence, which converges strongly to u in W ˚ γ 1 , p ( Ω ) .

In particular, the compactness property (A 4 ) follows from [21, Proposition 1.3] and the fact that, by Hölder’s inequality and the definition of the operator A p ,

⟨ A p ( u ) , v ⟩ ≤ ∥ u ∥ γ , p p − 1 ∥ v ∥ γ , p , ⟨ A p ( u ) , u ⟩ = ∥ u ∥ γ , p p for every u , v ∈ W ˚ γ 1 , p ( Ω ) .

Now we define a nondecreasing sequence { λ k } of eigenvalues of − Δ γ p by means of the cohomological index. This type of construction was introduced for the p -Laplacian by Perera [20] (see also [24]), and it is slightly different from the traditional one, based on the Krasnoselskii genus.

We recall that the Z 2 -cohomological index of Fadell and Rabinowitz [9] is defined as follows. Let W be a Banach space and let A denote the class of those subsets A of W \ { 0 } , which are symmetric in the sense that − A = A . For A ∈ A , let A ¯ = A ⁄ Z 2 be the quotient space of A with each u and − u identified, let f : A ¯ → R P ∞ be the classifying map of A ¯ , and let f * : H * ( R P ∞ ) → H * ( A ¯ ) be the induced homomorphism of the Alexander-Spanier cohomology rings. The cohomological index of A is defined by

i ( A ) = sup { m ≥ 1 ∣ f * ( ω n − ) ≠ 0 } , A ≠ ∅ , 0 , A = ∅ ,

where ω ∈ H 1 ( R P ∞ ) is the generator of the polynomial ring H * ( R P ∞ ) = Z 2 [ ω ] . The following proposition summarizes the basic properties of the cohomological index, see [9, Theorem 5.1].

Proposition 4.1

The index i : A → N ∪ { 0 , ∞ } satisfies the following properties.

Definiteness: i ( A ) = 0 if and only if A = ∅ ;
Monotonicity: if there is an odd continuous map from A to B, then i ( A ) ≤ i ( B ) . Thus, equality holds when the map is an odd homeomorphism;
Dimension: i ( A ) ≤ dim W ;
Continuity: if A is closed, then there is a closed neighborhood N ∈ A of A such that i ( N ) = i ( A ) . When A is compact, N may be chosen to be a δ -neighborhood N δ ( A ) = { u ∈ W ∣ dist ( u , A ) ≤ δ } ;
Subadditivity: if A and B are closed, then i ( A ∪ B ) ≤ i ( A ) + i ( B ) ;
Stability: if SA is the suspension of A ≠ ∅ , that is the quotient space of A × [ − 1 , 1 ] with A × { 1 } and A × { − 1 } collapsed to different points, then i ( S A ) = i ( A ) + 1 ;
Piercing property: if A , A 0 , and A 1 are closed, and φ : A × [ 0 , 1 ] → A 0 ∪ A 1 is a continuous map such that φ ( − u , t ) = − φ ( u , t ) for all ( u , t ) ∈ A × [ 0 , 1 ] , φ ( A × [ 0 , 1 ] ) is closed, φ ( A × { 0 } ) ⊂ A 0 and φ ( A × { 1 } ) ⊂ A 1 , then i ( φ ( A × [ 0 , 1 ] ) ∩ A 0 ∩ A 1 ) ≥ i ( A ) ;
Neighborhood of zero: if U is a bounded closed symmetric neighborhood of 0, then i ( ∂ U ) = dim W .

We define a C 1 -Finsler manifold ℳ by setting

ℳ = { u ∈ W ˚ γ 1 , p ( Ω ) ∣ ∥ u ∥ γ , p = 1 } .

For all k ∈ N , we denote by ℱ k the family of all closed, symmetric subsets M of ℳ such that i ( M ) ≥ k , and set

(15) λ k = inf M ∈ ℱ k sup u ∈ M Ψ ( u ) ,

where

Ψ ( u ) = 1 ∥ u ∥ p p , u ∈ ℳ \ { 0 } .

Given a ∈ R , we use the standard notation for the sublevels and superlevels of Ψ

Ψ a = { u ∈ W ˚ γ 1 , p ( Ω ) ∣ Ψ ( u ) ≥ a } , Ψ a = { u ∈ W ˚ γ 1 , p ( Ω ) ∣ Ψ ( u ) ≤ a } .

A straightforward application of [21, Theorem 4.6] to the operator A p yields the following spectral theory.

Proposition 4.2

The sequence { λ k } defined in (15) is a nondecreasing sequence of eigenvalues of − Δ γ p . Moreover,

the smallest eigenvalue, called the first eigenvalue, is
λ 1 = min u ≠ 0 ∥ u ∥ γ , p p ∥ u ∥ p p > 0 ;
we have i ( ℳ \ Ψ λ k ) < k ≤ i ( Ψ λ k ) . If λ k < λ < λ k + 1 , then
(16) i ( Ψ λ k ) = i ( ℳ \ Ψ λ ) = i ( Ψ λ ) = i ( ℳ \ Ψ λ k + 1 ) = k ;
λ k → + ∞ as k → + ∞ .

5 An abstract critical point theorem

Consider an even functional Φ of class C 1 on a Banach space W , let A * denote the class of symmetric subsets of W , let r > 0 and S r = { u ∈ W ∣ ∥ u ∥ = r } , let 0 < b ≤ + ∞ . Let Γ denote the group of odd homeomorphisms of W that are the identity outside Φ − 1 ( 0 , b ) . The pseudo-index of M ∈ A * related to S r and Γ is defined by

i * ( M ) = min γ ∈ Γ i ( γ ( M ) ∩ S r ) .

To obtain our result, we will apply the following critical point theorem, whose proof can be found in [23, Theorem 2.2].

Theorem 5.1

Let A 0 , B 0 be symmetric subsets of S 1 such that A 0 is compact, B 0 is closed, and

i ( A 0 ) ≥ k + m , i ( S 1 \ B 0 ) ≤ k

for some integers k ≥ 0 and m ≥ 1 . Assume that there exists R > r such that

sup Φ ( A ) ≤ 0 < inf Φ ( B ) , sup Φ ( X ) < b ,

where

A = { R u ∣ u ∈ A 0 } , B = { r u ∣ u ∈ B 0 } , X = { t u ∣ u ∈ A , 0 ≤ t ≤ 1 } .

For j = k + 1 , … , k + m , we set

A j * = { M ∈ A * ∣ M i s c o m p a c t a n d i * ( M ) ≥ j }

and

c j * = inf M ∈ A j * max u ∈ M Φ ( u ) .

Then

inf Φ ( B ) ≤ c k + 1 * ≤ … ≤ c k + m * ≤ sup Φ ( X ) ,

in particular, 0 < c j * < b . If, in addition, Φ satisfies the Palais-Smale condition for all levels c ∈ ( 0 , b ) ,^[2] then each c j * is a critical value of Φ , and there are m distinct pairs of associated critical points.

6 Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Solutions of problem (3) coincide with critical points of the C 1 -functional I γ : W ˚ γ 1 , p ( Ω ) → R defined by

I γ ( u ) = 1 p ∫ Ω ( ∣ ∇ γ u ∣ p − λ ∣ u ∣ p ) d z − 1 p γ * ∫ Ω ∣ u ∣ p γ * d z .

To apply Theorem 5.1 to the functional I λ , we need the functional I λ to satisfy the Palais-Smale conditions under a certain level, and we will use an argument of [12, Theorem 3.4].

Lemma 6.1

I γ satisfies the ( PS ) c conditions for all c < S N γ ⁄ p ⁄ N γ .

Proof

Let { u n } be a sequence in W ˚ γ 1 , p ( Ω ) , which satisfies the Palais-Smale conditions. First of all we claim that

(17) the sequence { u n } is bounded in W ˚ γ 1 , p ( Ω ) .

Indeed, for any n ∈ N , there exists k > 0 such that

(18) ∣ I γ ( u n ) ∣ ≤ k

and

I γ ′ ( u n ) , u n ‖ u n ‖ γ , p ≤ k ,

and so

(19) I γ ( u n ) − 1 p ⟨ I γ ′ ( u n ) , u n ⟩ ≤ k ( 1 + ∥ u n ∥ γ , p ) .

Furthermore,

I γ ( u n ) − 1 p ⟨ I γ ′ ( u n ) , u n ⟩ = 1 p − 1 p γ * ∥ u n ∥ p γ * p γ * = 1 N γ ∥ u n ∥ p γ * p γ *

so, thanks to (19), we obtain that for any n ∈ N

(20) ∥ u n ∥ p γ * p γ * ≤ k ˆ ( 1 + ∥ u n ∥ γ , p )

for a suitable positive constant k ˆ . By Hölder’s inequality, we obtain

∥ u n ∥ p p ≤ ∣ Ω ∣ p ⁄ N γ ∥ u n ∥ p γ * p ≤ k ˆ p ⁄ p γ * ∣ Ω ∣ p ⁄ N γ ( 1 + ∥ u n ∥ γ , p ) p ⁄ p γ * ,

that is,

(21) ∥ u n ∥ p p ≤ k ˜ ( 1 + ∥ u n ∥ γ , p ) ,

for a suitable k ˜ > 0 independent of j . By (18), (20), and (21), we have that

k ≥ I γ ( u n ) = 1 p ∥ u n ∥ γ , p p − λ p ∥ u n ∥ p p − 1 p γ * ∥ u n ∥ p γ * p γ * ≥ 1 p ∥ u n ∥ γ , p p − k ¯ ( 1 + ∥ u n ∥ γ , p )

for some constant k ¯ > 0 independent of n , so (17) is proved.

So, there exist a sequence, still denoted by { u n } , u ∈ W ˚ γ 1 , p ( Ω ) and T ∈ ( L p ′ ( Ω ) ) N such that u n ⇀ u weakly in W ˚ γ 1 , p ( Ω ) ∩ L p γ * ( Ω ) and strongly in L q ( Ω ) , 1 ≤ q < p γ * and ∣ ∇ γ u n ∣ p − 2 ∇ γ u n ⇀ T weakly in ( L p ′ ( Ω ) ) N . Moreover,

(22) − ∇ γ ⋅ T − ∣ u ∣ p γ * − 2 u − λ ∣ u ∣ p − 2 u = 0

in W γ − 1 , p ′ ( Ω ) .

From Theorem 3.3 (concentration-compactness), there exist two nonnegative bounded measures μ , ν in Ω , an at most countable family of points { z j } j ∈ J and positive numbers { ν j } j ∈ J such that

∣ ∇ γ u n ∣ p ⇀ ∗ μ , ∣ u n ∣ p γ * ⇀ ∗ ν

and

(23) ν = ∣ u ∣ p γ * + ∑ j ∈ J ν j δ z j ,

(24) μ ≥ ∣ ∇ γ u ∣ p + S ∑ j ∈ J ν j 1 − p ⁄ N γ δ z j .

Then, passing to the limit in the expression of I λ ( u n ) holds

(25) c = 1 p ∫ Ω d μ − 1 p γ * ∫ Ω d ν − λ p ∫ Ω ∣ u ∣ p d z .

Now, given φ ∈ C 1 ( Ω ¯ ) , testing (22) with the function u φ

(26) ∫ Ω ( u T ∇ γ φ + φ T ∇ γ u ) d z − ∫ Ω φ ∣ u ∣ p γ * d z − λ ∫ Ω φ ∣ u ∣ p d z = 0 ,

moreover, ⟨ I λ ′ ( u n ) , φ u n ⟩ → 0 as n → ∞ , that is,

∫ Ω u T ∇ γ φ d z + ∫ Ω φ d μ − ∫ Ω φ d ν − λ ∫ Ω φ ∣ u ∣ p d z = 0 .

This, combined with (23) and (26), implies

(27) ∫ Ω φ d μ = ∫ Ω φ T ∇ γ u d z + ∑ j ∈ J ν j φ ( z j ) .

Arguing as in the proof of Theorem 3.3, we pick a function φ ∈ C 1 ( Ω ¯ ) such that 0 ≤ φ ≤ 1 , φ ( 0 ) = 1 and supp φ = B ( 0 , 1 ) . For each ε > 0 , we apply equation (27) with φ j = φ ( z − z j ε ) , where j is fixed in J . This, combined with (24), gives

ν j ≥ ∫ Ω φ j ( ∣ ∇ γ u ∣ p − T ∇ γ u ) d z + S ν j 1 − p ⁄ N γ .

Letting ε → 0 , we obtain

ν j ≥ S N γ ⁄ p , ∀ j ∈ J .

∫ Ω T ∇ γ u d z = ∫ Ω ∣ u ∣ p γ * d z + λ ∫ Ω ∣ u ∣ p d z ,

combined with (25) and (27) considering φ ≡ 1 gives

c = 1 p − 1 p γ * ∑ j ∈ J ν j + 1 p − 1 p γ * ∫ Ω ∣ u ∣ p γ * d z ≥ 1 N γ S N γ ⁄ p ,

which is in contradiction with the hypothesis c < S N γ ⁄ p ⁄ N γ . We deduce that J is empty and

lim n → ∞ ∫ Ω ∣ u n ∣ p γ * d z = ∫ Ω ∣ u ∣ p γ * d z ,

and thus,

(28) u n → u strongly in L p γ * ( Ω ) .

A straightforward computation shows that the sequence { A p ( u n ) } is a Cauchy sequence in W γ − 1 , p ′ ( Ω ) . In fact, A p ( u n ) = I λ ′ ( u n ) + λ ∣ u n ∣ p − 2 u n + ∣ u n ∣ p γ * − 2 u n and the claim follows from (28) and the Palais-Smale condition. Now, a direct application of [25, Eq (2.2)] implies

⟨ A p ( u n ) − A p ( u m ) , u n − u m ⟩ ≥ c 1 ∥ u n − u m ∥ γ , p p if p > 2 , c 2 M p − 2 ∥ u n − u m ∥ γ , p 2 if 1 < p ≤ 2 ,

where c 1 = c 1 ( N , γ , p ) , c 2 = c 2 ( N , γ , p , Ω ) and M = max { ∥ u n ∥ γ , p , ∥ u m ∥ γ , p } . So

∥ u n − u m ∥ γ , p ≤ c 1 1 1 − p ∥ A p ( u n ) − A p ( u m ) ∥ W γ − 1 , p ′ ( Ω ) 1 p − 1 if p > 2 , c 2 − 1 M 2 − p ∥ A p ( u n ) − A p ( u m ) ∥ W γ − 1 , p ′ ( Ω ) if 1 < p ≤ 2 ,

so we deduce that u n → u strongly in W ˚ γ 1 , p ( Ω ) .□

If λ k + m < λ k + m + 1 , then i ( Ψ λ k + m ) = k + m by (16). We now construct a symmetric subset A 0 of Ψ λ k + m with the same cohomological index. We need some further properties of the operator A p introduced in Section 4.

Lemma 6.2

The operator A p is strictly monotone, i.e.,

⟨ A p ( u ) − A p ( v ) , u − v ⟩ > 0

for all u ≠ v in W ˚ γ 1 , p ( Ω ) .

Proof

To apply [21, Lemma 6.3], it suffices to show that

⟨ A p ( u ) , v ⟩ ≤ ∥ u ∥ γ , p p − 1 ∥ v ∥ γ , p

for every u , v ∈ W ˚ γ 1 , p ( Ω ) , and the equality holds if and only if α u = β v for some α ≥ 0 , β ≥ 0 , not both zero. By Hölder’s inequality,

⟨ A p ( u ) , v ⟩ = ∫ Ω ∣ ∇ γ u ∣ p − 2 ∇ γ u ∇ γ v d z ≤ ∫ Ω ∣ ∇ γ u ∣ p − 1 ∣ ∇ γ v ∣ ≤ ∥ u ∥ γ , p p − 1 ∥ v ∥ γ , p .

Clearly, equality holds throughout if and only if α u = β v for some α ≥ 0 , β ≥ 0 , not both zero.

Conversely, if ⟨ A p ( u ) , v ⟩ = ∥ u ∥ γ , p p − 1 ∥ v ∥ γ , p , equality holds in both inequalities. The equality in the Hölder’s inequality give

α ∣ ∇ γ u ∣ = β ∣ ∇ γ v ∣ a.e.

for some α , β ≥ 0 , not both zero, and the equality in the Schwartz inequality gives

α ∇ γ u = β ∇ γ v a.e.,

so α u = β v .□

Lemma 6.3

For each w ∈ L p ( Ω ) , the problem

− Δ γ p u = ∣ w ∣ p − 2 w i n Ω , u = 0 o n ∂ Ω ,

has a unique weak solution u ∈ W ˚ γ 1 , p ( Ω ) . Moreover, the map J : L p ( Ω ) → W ˚ γ 1 , p ( Ω ) , w ↦ u is continuous.

Proof

The existence of a solution follows from a straightforward minimization procedure, and uniqueness is immediate from the strict monotonicity of the operator A p . Let w j → w in L p ( Ω ) and let u j = J ( w j ) , u = J ( w ) . By definition,

⟨ A p ( u j ) , v ⟩ = ∫ Ω ∣ w j ∣ p − 2 w j v d z for all v ∈ W ˚ γ 1 , p ( Ω ) .

Testing with v = u j gives

∥ u j ∥ γ , p p ≤ ∥ w j ∥ p p − 1 ∥ u j ∥ p p

by Hölder’s inequality, which together with the continuity of the embedding W ˚ γ 1 , p ( Ω ) ↪ L p ( Ω ) shows that { u j } is bounded in W ˚ γ 1 , p ( Ω ) . By reflexivity, up to a subsequence u j ⇀ u * in W ˚ γ 1 , p ( Ω ) and strongly in L p ( Ω ) . By the continuity of the Nemitskii operator,

∣ w j ∣ p − 2 w j → ∣ w ∣ p − 2 w

strongly in L p ′ ( Ω ) . Therefore, A p ( u j ) → A p ( u ) , or

⟨ A p ( u j ) , v ⟩ = ∫ Ω ∣ w j ∣ p − 2 w j v d z → ∫ Ω ∣ w ∣ p − 2 w v d z = ⟨ A p ( u ) , v ⟩

for every v ∈ W ˚ γ 1 , p ( Ω ) . Hence, J ( w ) = u . Now,

⟨ A p ( u j ) , u j − u * ⟩ = ∫ Ω ∣ w j ∣ p − 2 w j ( u j − u * ) d z → 0

since { ∣ w j ∣ p − 2 w j } is bounded in L p ′ ( Ω ) and u j → u * strongly in L p ( Ω ) . Since A p is of type (S), up to another subsequence u j → u * strongly in W ˚ γ 1 , p ( Ω ) . It follows easily that J ( u * ) = w , and by uniqueness u * = u . A standard argument shows now that the whole sequence { u j } converges to u , since each subsequence converges to the same limit u .□

Proposition 6.4

If λ ℓ < λ ℓ + 1 , then Ψ λ ℓ has a compact symmetric subset A 0 with i ( A 0 ) = ℓ .

Proof

Let

π p ( u ) = u ∥ u ∥ p , u ∈ W ˚ γ 1 , p ( Ω ) \ { 0 }

be the radial projection onto ℳ p = { u ∈ W ˚ γ 1 , p ( Ω ) ∣ ∥ u ∥ p = 1 } , and let

A = π p ( Ψ λ ℓ ) = { w ∈ ℳ p ∣ ∥ u ∥ γ , p p ≤ λ ℓ } ,

which is compact in L p ( Ω ) since the embedding W ˚ γ 1 , p ( Ω ) ↪ L p ( Ω ) is compact.

Then i ( A ) = i ( Ψ λ ℓ ) = ℓ by (i 2 ) of Proposition 4.1 and (16). For w ∈ A , let u = J ( w ) , where J is the map defined in Lemma 6.3, so

⟨ A p ( u ) , v ⟩ = ∫ Ω ∣ w ∣ p − 2 w v d z , ∀ v ∈ W ˚ γ 1 , p ( Ω ) .

Testing with v = u , w and using Hölder’s inequality give

∥ u ∥ γ , p ≤ ∥ w ∥ p p − 1 ∥ u ∥ p = ∥ u ∥ p , 1 = ⟨ A p ( u ) , w ⟩ ≤ ∥ u ∥ γ , p p − 1 ∥ w ∥ γ , p ,

∥ π p ( u ) ∥ γ , p = ∥ u ∥ γ , p ∥ u ∥ p ≤ ∥ w ∥ γ , p ,

and hence, π p ( u ) ∈ A .

Let J ˜ = π p ∘ J and let A ˜ = J ˜ ( A ) ⊂ A . Since J ˜ is an odd continuous map from L p ( Ω ) to W ˚ γ 1 , p ( Ω ) and A is compact in L p ( Ω ) , then A ˜ is a compact set and i ( A ˜ ) = i ( A ) = ℓ by (i 2 ) of Proposition 4.1. Let

π ( u ) = u ∥ u ∥ γ , p , u ∈ W ˚ γ 1 , p ( Ω ) \ { 0 } ,

be the radial projection onto ℳ = { u ∈ W ˚ γ 1 , p ( Ω ) ∣ ∥ u ∥ γ , p = 1 } , and let A 0 = π ( A ˜ ) . Then A 0 ⊂ Ψ λ ℓ is compact and i ( A 0 ) = i ( A ˜ ) = ℓ .□

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1

We only give the proof of (2). The proof of (1) is similar and simpler. By Lemma 6.1, I λ satisfies the (PS) c conditions for all c < S N γ ⁄ p ⁄ N γ , so we apply Theorem 5.1 with b = S N γ ⁄ p ⁄ N γ . By Proposition 6.4, Ψ λ k + m has a compact symmetric subset A 0 with

i ( A 0 ) = k + m .

We take B 0 = Ψ λ k + 1 , so that

i ( S 1 \ B 0 ) = k

by (16). Let R > r > 0 and let A , B , and X be as in Theorem 5.1. For u ∈ B 0 ,

I λ ( r u ) ≥ r p p 1 − λ λ k + 1 − r p γ * p γ * S p γ * ⁄ p

by the definition (7) of the constant S . Since λ < λ k + 1 and p γ * > p , it follows that inf I λ ( B ) > 0 if r sufficiently small. For u ∈ A 0 ⊂ Ψ λ k + 1 ,

I λ ( R u ) ≤ R p p 1 − λ λ k + 1 − R p γ * p γ * ∣ Ω ∣ p γ * ⁄ N λ k + 1 p γ * ⁄ p

by Hölder’s inequality, so there exists R > r such that I λ ≤ 0 on A . For u ∈ X ,

I λ ( u ) ≤ λ k + 1 − λ p ∫ Ω ∣ u ∣ p d z − 1 p γ * ∣ Ω ∣ p γ * ⁄ N γ ∫ Ω ∣ u ∣ p d z p γ * ⁄ p ≤ sup ρ ≥ 0 ( λ k + 1 − λ ) ρ p − ρ p γ * ⁄ p p γ * ∣ Ω ∣ p γ * ⁄ N γ = ∣ Ω ∣ N γ ( λ k + 1 − λ ) N γ ⁄ p .

So,

sup I λ ( X ) ≤ ∣ Ω ∣ N γ ( λ k + 1 − λ ) N γ ⁄ p < S N γ ⁄ p N γ

by (5). Theorem 5.1 now gives m distinct pairs of (nontrivial) critical points ± u j λ , j = 1 , … , m , of I λ such that

0 < I λ ( u j λ ) ≤ ∣ Ω ∣ N γ ( λ k + 1 − λ ) N γ ⁄ p → 0 as λ ↗ λ k + 1 .

Then

∥ u j λ ∥ p γ * p γ * = N γ I λ ( u j λ ) − 1 p ⟨ I λ ′ ( u j λ ) , u j λ ⟩ = N γ I λ ( u j λ ) → 0

as λ ↗ λ k + 1 , and hence, u j λ → 0 in L p ( Ω ) also by Hölder’s inequality, so

∥ u j λ ∥ γ , p p = p I λ ( u j λ ) + λ ∥ u j λ ∥ p p + p p γ * ∥ u j λ ∥ p γ * p γ * → 0 ,

as λ ↗ λ k + 1 , this concludes the proof.□

Acknowledgments

This work has been funded by the European Union - NextGenerationEU within the framework of PNRR Mission 4 - Component 2 - Investment 1.1 under the Italian Ministry of University and Research (MUR) program PRIN 2022 - grant number 2022BCFHN2 - “Advanced theoretical aspects in PDEs and their applications” - CUP: H53D23001960006 and the authors were partially supported by GNAMPA 2024: “Aspetti geometrici e analitici di alcuni problemi locali e nonlocali in mancanza di compattezza.”

Funding information: PRIN 2022 – grant number 2022BCFHN2 – “Advanced theoretical aspects in PDEs and their applications” – CUP: H53D23001960006.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2025-01-18

Revised: 2025-04-23

Accepted: 2025-04-30

Published Online: 2025-08-08

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Keywords for this article

p-Grushin operator; critical exponent

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