Existence of nontrivial solutions for critical Kirchhoff-Poisson systems in the Heisenberg group

Patrizia Pucci; Yiwei Ye

doi:10.1515/ans-2022-0018

Artikel Open Access

Existence of nontrivial solutions for critical Kirchhoff-Poisson systems in the Heisenberg group

Patrizia Pucci und Yiwei Ye

Veröffentlicht/Copyright: 11. August 2022

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Aus der Zeitschrift Advanced Nonlinear Studies Band 22 Heft 1

Abstract

This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system

− M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u ln ∣ u ∣ 2 + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω ,

where Δ H is the Kohn-Laplacian operator in the first Heisenberg group H 1 , Ω is a smooth bounded domain of H 1 , q ∈ ( 2 θ , 4 ) , μ ∈ R , and λ > 0 are some real parameters. Under suitable assumptions on the Kirchhoff function M , which cover the degenerate case, we prove the existence of nontrivial solutions for the above problem when λ > 0 is sufficiently large. Moreover, our results are new even in the Euclidean case.

Keywords: Heisenberg group; Kirchhoff-Poisson system; critical growth; logarithmic nonlinearity; concentration-compactness principle

MSC 2010: Primary: 35A15; 35J60; 35R03; Secondary: 47J20

1 Introduction and main results

Consider the following Kirchhoff-Poisson system with logarithmic and critical nonlinearity in the Heisenberg group:

(1.1) − M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u ln ∣ u ∣ 2 + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω ,

where Δ H is the Kohn-Laplacian operator in the first Heisenberg group H 1 , Ω ⊂ H 1 is a smooth bounded domain, q ∈ ( 2 θ , 4 ) and θ is given by condition ( M 2 ) below, and μ ∈ R and λ > 0 are real parameters. Let us put, for simplicity, R 0 + = [ 0 , ∞ ) and R + = ( 0 , ∞ ) . Concerning the Kirchhoff term M , we assume that M ∈ C ( R 0 + , R 0 + ) satisfies the following:

For any τ > 0 , there exists m 0 = m 0 ( τ ) > 0 such that M ( t ) ≥ m 0 for t ≥ τ .
There exists θ ∈ [ 1 , 2 ) such that θ M ^ ( t ) ≥ M ( t ) t for all t ≥ 0 , where M ^ ( t ) = ∫ 0 t M ( s ) d s .
There exists m 1 > 0 such that M ( t ) ≥ m 1 t θ − 1 for all t ∈ R + and M ( 0 ) = 0 .

A typical example is given by

M ( t ) = a + b t θ − 1 , a , b ≥ 0 , a + b > 0 , θ ≥ 1 .

When M is of this type, problem (1.1) is called nondegenerate if a > 0 , and degenerate if a = 0 .

In recent years, geometric analysis in the Heisenberg group has become one of the most active and exciting research fields. This is because the Heisenberg group plays a crucial role in several branches of mathematics, such as representation theory, harmonic analysis, complex variables, quantum mechanics, and partial differential equations, see [6,8,13,14,21,24]. For example, [24] deals with multiplicity for entire solutions to a quasilinear equation in the Heisenberg group H n , depending on a real parameter λ . It proves that for any λ > 0 , there exist infinitely many solutions ( u k ) k with negative critical values that tend to zero as k → ∞ . In [21], the authors study the existence of entire solutions for the critical quasilinear elliptic systems in the Heisenberg group, involving ( p , q ) operators. The proof relies on the adaptation of Lions’ concentration-compactness principle in the vectorial Heisenberg context and variational methods. See also [4,22,23] for the related results.

In the Euclidean setting, the existence and multiplicity of solutions for the nonlocal problems (i.e., Kirchhoff-type problems, Schrödinger-Poisson systems, and so on) have been widely studied, and many recent interesting results are obtained. We just quote, for example, [7,15,26,27]. Xiang et al. [26] consider the fractional nondegenerate Kirchhoff equations with logarithmic nonlinearity

M ( [ u ] s , p p ) ( − Δ ) p s u = h ( x ) ∣ u ∣ θ p − 2 u ln ∣ u ∣ + λ ∣ u ∣ q − 2 u , x ∈ Ω , u = 0 , x ∈ R N \ Ω ,

where s ∈ ( 0 , 1 ) , p ∈ ( 1 , N / s ) , θ ∈ ( 1 , p s ∗ / p ) , Ω is a bounded domain with Lipschitz boundary of R N , and h ∈ C ( Ω ¯ ) is a sign-changing function. By applying the Nehari manifold approach, they prove the existence of two local least energy solutions for any exponent q ∈ ( 1 , θ p ) and λ > 0 small enough. However, the existence and multiplicity results of solutions of Kirchhoff-Poisson systems in the Heisenberg group are very few, see [2,16, 17,18]. The Kirchhoff-Poisson system with critical nonlinearity of the form

− a − b ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω

has been studied by Liu et al. [18], where the authors discussed the cases q ∈ ( 1 , 2 ) and q ∈ ( 2 , 4 ) and proved the existence and multiplicity results under suitable assumptions on μ and λ . Very recently, Liang and Pucci [16] have investigated the following Kirchhoff-Poisson system in the nondegenerate case

− M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + ϕ ∣ u ∣ q − 2 u = h ( ξ , u ) + λ ∣ u ∣ 2 u in Ω , − Δ H ϕ = ∣ u ∣ q in Ω , u = ϕ = 0 on ∂ Ω .

Besides some other conditions, they assume that q ∈ ( 1 , 2 ) and M ( t ) ≥ m 0 > 0 for all t ∈ R 0 + , and that there exists θ ∈ ( 2 q , 4 ) such that 0 < θ H ( ξ , t ) ≤ h ( ξ , t ) t for x ∈ Ω and ∣ t ∣ ≥ T , and they prove a multiplicity result when λ is sufficiently small. To the best of our knowledge, there are no results concerning the existence and multiplicity of solutions of the Kirchhoff-Poisson system (1.1) with logarithmic and critical nonlinearities in the Heisenberg group, even in the Euclidean case.

Inspired by the aforementioned works, we are interested in the study of the combined effects of logarithmic and critical nonlinearities for system (1.1) in the Heisenberg group. To this aim, let us recall that in [10], Folland and Stein introduced the Hilbert space S 0 1 ( Ω ) as the closure of C 0 ∞ ( Ω ) under the inner product ⟨ u , v ⟩ ≔ ∫ Ω ∇ H u ∇ H v d ξ , with Hilbertian corresponding norm

‖ u ‖ = ‖ u ‖ S 0 1 ( Ω ) = ∫ Ω ∣ ∇ H u ∣ 2 d ξ 1 2 .

The embedding S 0 1 ( Ω ) ↪ L s ( Ω ) is continuous for s ∈ [ 1 , Q ∗ ] , and the embedding is compact if and only if s ∈ [ 1 , Q ∗ ) , where Q ∗ ≔ 2 Q Q − 2 = 4 is the critical exponent in H 1 . The best Sobolev constant

(1.2) S = inf u ∈ S 0 1 ( H 1 ) \ { 0 } ∫ H 1 ∣ ∇ H u ∣ 2 d ξ ∫ H 1 ∣ u ∣ 4 d ξ 1 2

is achieved by the C ∞ function U ( x , y , t ) = c 0 [ ( 1 + x 2 + y 2 ) 2 + t 2 ] 1 2 , where c 0 > 0 is a constant (see [12]).

The main result of the article is the following.

Theorem 1.1

Assume that ( M 1 )–( M 3 ) are satisfied and μ < S ∣ Ω ∣ − 1 2 , where S is the best Sobolev constant given by (1.2). Then there exists λ ∗ > 0 such that problem (1.1) has a nontrivial solution for any λ > λ ∗ .

Remark 1.1

The features of Theorem 1.1 are as follows:

the presence of the logarithmic term;
the presence of the critical nonlinearity, which contributes to the lack of compactness; and
the fact that the result includes the degenerate case, which corresponds to the Kirchhoff function M vanishing at zero.

We point out that the degenerate case is rather appealing, not only from a mathematical point of view but also in applications. 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. It is treated in famous well-known articles in Kirchhoff theory, see [9]. In addition, let us note that although the Kohn-Laplacian Δ H and the classical Laplacian Δ have similar properties, the similarities may be misleading (see [11]). Moreover, the critical exponent Q ∗ = 4 in H 1 , while 2 ∗ = 6 in R 3 , which causes some obstacles in proving the compactness. In order to overcome these difficulties, we use the concentration-compactness principle in the Heisenberg group and carefully analyze the competing nonlinear terms to prove that the ( P S ) c condition holds at suitable levels of c .

The article is organized as follows. In Section 2, we present some preliminaries on the Heisenberg group functional setting and prove the local Palais-Smale condition. Section 3 is devoted to the proof of Theorem 1.1.

2 Preliminaries

We briefly recall some definitions and notations on the Heisenberg group. For a complete treatment, we refer to [5,11].

Let H 1 be the Heisenberg group of topological dimension 3, that is, the Lie group where underlying manifold is R 3 , endowed with the nonAbelian law

τ : H 1 → H 1 , τ ξ ( ξ ′ ) = ξ ∘ ξ ′ ,

where

ξ ∘ ξ ′ = ( x + x ′ , y + y ′ , t + t ′ + 2 ( x ′ y − x y ′ ) )

for all ξ , ξ ′ ∈ H 1 , with ξ = ( x , y , t ) and ξ ′ = ( x ′ , y ′ , t ′ ) . The inverse is given by ξ − 1 = − ξ , and hence ( ξ ∘ ξ ′ ) − 1 = ( ξ ′ ) − 1 ∘ ξ − 1 . Consider the family of dilations on H 1 defined by

δ s ( ξ ) = ( s x , s y , s 2 t ) , ∀ ξ ∈ H 1 ,

so δ s ( ξ ∘ ξ ′ ) = δ s ( ξ ) ∘ δ s ( ξ ′ ) (see [21]). It is easy to check that the Jacobian determinate of the dilatations δ s : H 1 → H 1 is a constant and equals to s 4 . As a result, the number Q = 4 is the homogeneous dimension of H 1 . The Haar measure on H 1 coincides with the Lebesgue measure on R 3 . It is invariant under left translations and Q -homogeneous with respect to dilations. Then

∣ B H ( ξ 0 , r ) ∣ = ω Q r Q ,

where B H ( ξ 0 , r ) is the Heisenberg ball of radius r centered at ξ 0 , i.e.,

B H ( ξ 0 , r ) = { ξ ∈ H 1 : d H ( ξ 0 , ξ ) < r }

and ω Q = ∣ B H ( 0 , 1 ) ∣ .

The Kohn-Laplacian Δ H on H 1 is defined as

Δ H u = div H ( ∇ H u ) ,

where ∇ H u = ( X u , Y u ) . Indeed, the vector fields

X = ∂ ∂ x + 2 y ∂ ∂ t , Y = ∂ ∂ y − 2 x ∂ ∂ t , and T = ∂ ∂ t

constitute a basis for the Lie algebra of left-invariant vector fields on H 1 . It is well known that Δ H is a degenerate elliptic operator, and the Bony maximum principle is satisfied (see [3]).

First, we consider the problem

(2.1) − Δ H ϕ = u 2 in Ω , ϕ = 0 on ∂ Ω .

It follows from the Lax-Milgram theorem that for every u ∈ S 0 1 ( Ω ) , problem (2.1) has a unique solution ϕ u ∈ S 0 1 ( Ω ) . Moreover, by the maximum principle, ϕ u ≥ 0 and ϕ u > 0 if u ≠ 0 . We give some properties of the solution ϕ u , and the detailed proof can be found in [2].

Proposition 2.1

(see [2]) Let u ∈ S 0 1 ( Ω ) be fixed. The corresponding solution ϕ u ∈ S 0 1 ( Ω ) of problem (2.1), has the properties

ϕ u ≥ 0 and ϕ t u = t 2 ϕ u for all t > 0 ;
∫ Ω ∣ ∇ H ϕ u ∣ 2 d ξ = ∫ Ω ϕ u u 2 d ξ ≤ S − 1 ‖ u ‖ 8 3 4 ≤ S − 1 ∣ Ω ∣ 1 2 ∫ Ω ∣ u ∣ 4 d ξ ;
let u n ⇀ u in S 0 1 ( Ω ) , then ϕ u n ⇀ ϕ u in S 0 1 ( Ω ) and
lim n → ∞ ∫ Ω ϕ u n u n v d ξ = ∫ Ω ϕ u u v d ξ for all v ∈ S 0 1 ( Ω ) .

On S 0 1 ( Ω ) , we define the functional

J λ ( u ) = 1 2 M ^ ( ‖ u ‖ 2 ) + μ 4 ∫ Ω ϕ u ∣ u ∣ 2 d ξ + λ ∫ Ω 2 q 2 ∣ u ∣ q − 1 q ∣ u ∣ q ln ∣ u ∣ 2 d ξ − 1 4 ∫ Ω ∣ u ∣ 4 d ξ .

From Proposition 2.1, it is easy to check that the functional J λ is well defined in S 0 1 ( Ω ) . Moreover, J λ ∈ C 1 ( S 0 1 ( Ω ) , R ) and

⟨ J λ ′ ( u ) , v ⟩ = M ( ‖ u ‖ 2 ) ⟨ u , v ⟩ + μ ∫ Ω ϕ u u v d ξ − λ ∫ Ω ∣ u ∣ q − 2 ln ∣ u ∣ 2 u v d ξ − ∫ Ω ∣ u ∣ 2 u v d ξ

for all u , v ∈ S 0 1 ( Ω ) . The critical points of J λ correspond to the solutions of problem (1.1).

Since for all q ∈ ( 2 θ , 4 ) and r ∈ ( q , 4 )

lim t → 0 ∣ t ∣ q − 1 ln ∣ t ∣ 2 ∣ t ∣ 2 θ − 1 = 0 and lim ∣ t ∣ → ∞ ∣ t ∣ q − 1 ln ∣ t ∣ 2 ∣ t ∣ r − 1 = 0 ,

for any ε > 0 , there exists C ε > 0 such that

(2.2) ∣ t ∣ q − 1 ln ∣ t ∣ 2 ≤ ε ∣ t ∣ 2 θ − 1 + C ε ∣ t ∣ r − 1 .

Hence, if u n ⇀ u in S 0 1 ( Ω ) , then the Vitali convergence theorem implies that

(2.3) ∫ Ω ∣ u n ∣ q ln ∣ u n ∣ 2 d ξ → ∫ Ω ∣ u ∣ q ln ∣ u ∣ 2 d ξ as n → ∞ .

Given c ∈ R , we say that a sequence ( u n ) n ⊂ S 0 1 ( Ω ) is a ( P S ) c sequence for the functional J λ at the level c if J λ ( u n ) → c and J λ ′ ( u n ) → 0 as n → ∞ . Moreover, J λ is said to satisfy ( P S ) c condition at the level c if any ( P S ) c sequence possesses a strongly convergent subsequence in S 0 1 ( Ω ) . Let us prove

Lemma 2.1

Assume that conditions ( M 1 )–( M 3 ) hold and μ < S ∣ Ω ∣ − 1 2 . Then J λ satisfies the ( P S ) c condition at any

c ∈ I , I ≔ 0 , 1 2 θ − 1 q m 1 2 2 − θ S 2 θ 2 − θ .

Proof

Let c be in I and let ( u n ) n be a ( P S ) c sequence of J λ , i.e.,

(2.4) J λ ( u n ) → c and J λ ′ ( u n ) → 0

as n → ∞ . Proposition 2.1 gives that

(2.5) ∫ Ω ∣ u ∣ 4 d ξ − μ ∫ Ω ϕ u ∣ u ∣ 2 d ξ ≥ ∫ Ω ∣ u ∣ 4 d ξ , if μ ≤ 0 , 1 − μ S − 1 ∣ Ω ∣ 1 2 ∫ Ω ∣ u ∣ 4 d ξ , if 0 < μ < S ∣ Ω ∣ − 1 2 ,

so that

(2.6) c + o ( 1 ) ‖ u n ‖ = J λ ( u n ) − 1 q ⟨ J λ ′ ( u n ) , u n ⟩ ≥ 1 2 θ − 1 q M ( ‖ u ‖ 2 ) ‖ u ‖ 2 + 2 λ q 2 ∫ Ω ∣ u n ∣ q d ξ + 1 q − 1 4 ∫ Ω ∣ u n ∣ 4 d ξ − μ ∫ Ω ϕ u n ∣ u n ∣ 2 d ξ ≥ 1 2 θ − 1 q m 1 ‖ u n ‖ 2 θ ,

by ( M 2 )–( M 3 ) and the fact that q ∈ ( 2 θ , 4 ) . This implies that ( u n ) n is bounded in S 0 1 ( Ω ) . Hence by [20] and Proposition 2.1, passing eventually to a subsequence, we may assume that for some u ∈ S 0 1 ( Ω )

(2.7) u n ⇀ u in S 0 1 ( Ω ) , ϕ u n ⇀ ϕ u in S 0 1 ( Ω ) , u n → u in L s ( Ω ) , with 1 ≤ s < 4 , u n → u a.e. in Ω .

Now we claim that

(2.8) ‖ u n ‖ 2 → ‖ u ‖ 2 as n → ∞ ,

implying that u n → u in S 0 1 ( Ω ) as n → ∞ .

In fact, it follows from the concentration-compactness principle on the Heisenberg group (see [25, Lemma 3.5]) that there exist an at most countable set of distinct points { x j } j ∈ Λ ⊂ Ω , nonnegative numbers { ω j } j ∈ Λ , { ν j } j ∈ Λ , and two positive Radon measures ω and ν in H 1 , with support in Ω , such that

(2.9) ∣ ∇ H u n ∣ 2 d ξ ⇀ ∗ d ω and ∣ u n ∣ 4 d ξ ⇀ ∗ d ν in ℳ ( H 1 ) , d ω ≥ ∣ ∇ H u ∣ 2 d ξ + ∑ j ∈ Λ ω j δ x j , d ν = ∣ u ∣ 4 d ξ + ∑ j ∈ Λ ν j δ x j ,

and

(2.10) ω j ≥ S ν j 1 2 .

In order to prove (2.8), we proceed by steps.

Step 1. Fixed j ∈ Λ . Then, either ω j = 0 or

(2.11) ω j ≥ ( m 1 S 2 ) 1 2 − θ .

For ε > 0 small, we set ψ j , ε ∈ C 0 ∞ ( B H ( ξ j , ε ) ) such that 0 ≤ ψ j , ε ( ξ ) ≤ 1 , ψ j , ε ( ξ ) = 1 in B H ( ξ j , ε / 2 ) , ψ j , ε ( ξ ) = 0 in Ω \ B ( ξ j , ε ) , and ∣ ∇ H ψ j , ε ∣ ≤ 2 / ε . Clearly, ( u n ψ j , ε ) n is bounded in S 0 1 ( Ω ) , and so (2.4) implies that

⟨ J λ ′ ( u n ) , u n ψ j , ε ⟩ → 0 as n → ∞ ,

that is,

(2.12) M ( ‖ u n ‖ 2 ) ∫ Ω ∣ ∇ H u n ∣ 2 ψ j , ε d ξ + ∫ Ω u n ∇ H u n ∇ H ψ j , ε d ξ + μ ∫ Ω ϕ u n ∣ u n ∣ 2 ψ j , ε d ξ = λ ∫ Ω ∣ u n ∣ q ln ∣ u n ∣ 2 ψ j , ε d ξ + ∫ Ω ∣ u n ∣ 4 ψ j , ε d ξ + o ( 1 ) .

By the dominated convergence theorem, we obtain that

∫ B H ( x j , ε ) ∣ u n ∣ q ln ∣ u n ∣ 2 ψ j , ε d ξ → ∫ B H ( x j , ε ) ∣ u ∣ q ln ∣ u ∣ 2 ψ j , ε d ξ

as n → ∞ , and then, by sending ε → 0 ,

(2.13) lim ε → 0 lim n → ∞ ∫ B H ( x j , ε ) ∣ u n ∣ q ln ∣ u n ∣ 2 ψ j , ε d ξ = 0 .

Proposition 2.1(iii) gives

lim n → ∞ ∫ Ω ϕ u n u n u d ξ = ∫ Ω ϕ u ∣ u ∣ 2 d ξ .

Moreover, by (2.7),

∫ Ω ( ϕ u n ∣ u n ∣ 2 − ϕ u u n u ) d ξ ≤ ∫ Ω ∣ ϕ u n ∣ ∣ u n ∣ ∣ u n − u ∣ d ξ ≤ ∣ ϕ u n ∣ 4 ∣ u n ∣ 8 / 3 ∣ u n − u ∣ 8 / 3 → 0 .

Consequently, the last two limits give at once

lim n → ∞ ∫ Ω ϕ u n ∣ u n ∣ 2 d ξ = ∫ Ω ϕ u ∣ u ∣ 2 d ξ .

Thus,

(2.14) lim ε → 0 lim n → ∞ ∫ Ω ϕ u n ∣ u n ∣ 2 ψ j , ε d ξ = lim ε → 0 ∫ B H ( x j , ε ) ϕ u ∣ u ∣ 2 ψ j , ε d ξ = 0 .

Moreover, applying the Heisenberg polar coordinates (see [19]), we deduce that

∫ B H ( x j , ε ) d ξ = ∫ B H ( 0 , ε ) d ξ = ∣ B H ( 0 , 1 ) ∣ ε 4 ,

and then, using the Hölder inequality,

(2.15) lim ε → 0 lim n → ∞ ∫ Ω u n ∇ H u n ∇ H ψ j , ε d ξ ≤ lim ε → 0 lim n → ∞ ∫ B ( x j , ε ) ∣ ∇ H u n ∣ 2 d ξ 1 2 ∫ B ( x j , ε ) ∣ u n ∣ 2 ∣ ∇ H ψ j , ε ∣ 2 d ξ 1 2 ≤ C lim ε → 0 ∫ Ω ∣ u ∣ 2 ∣ ∇ H ψ j , ε ∣ 2 d ξ 1 2 ≤ C lim ε → 0 ∫ B ( x j , ε ) ∣ u ∣ 4 d ξ 1 4 ∫ B ( x j , ε ) ∣ ∇ H ψ j , ε ∣ 4 d ξ 1 4 = 0 .

Hence, combining (2.12)–(2.15) and condition ( M 3 ), we obtain the key inequality

ν j ≥ m 1 ω j θ ,

which, jointly with (2.10), yields that either ω j = 0 or ω j verifies (2.11).

Step 2. Estimate (2.11) cannot occur, and hence ω j = 0 for all j .

Indeed, if (2.11) holds, then by (2.6),

c = lim n → ∞ J λ ( u n ) − 1 q ⟨ J λ ′ ( u n ) , u n ⟩ ≥ lim n → ∞ 1 2 θ − 1 q m 1 ∫ Ω ∣ ∇ H u n ∣ 2 ψ j , ε d ξ θ = 1 2 θ − 1 q m 1 ∫ Ω ψ j , ε d μ θ ,

and so, letting ε → 0 ,

c ≥ 1 2 θ − 1 q m 1 ω j θ ≥ 1 2 θ − 1 q m 1 2 2 − θ S 2 θ 2 − θ ∉ I ,

which is impossible.

Step 3. Claim (2.8) holds true.

Since j is arbitrary in Step 1, we deduce that ω j = 0 for all j ∈ Λ . As a consequence, from (2.9), it follows that

(2.16) ∫ Ω ∣ u n ∣ 4 d ξ → ∫ Ω ∣ u ∣ 4 d ξ as n → ∞ .

Let lim n → ∞ ‖ u n ‖ 2 = A . If A = 0 , i.e., u n → 0 in S 0 1 ( Ω ) , then using (2.7), (2.3), and the fact M ( 0 ) = 0 , we see that

c + o ( 1 ) = J λ ( u n ) − 1 4 ⟨ J λ ′ ( u n ) , u n ⟩ = 1 2 M ^ ( ‖ u n ‖ 2 ) − 1 4 M ( ‖ u n ‖ 2 ) ‖ u n ‖ 2 + 2 λ q 2 ∫ Ω ∣ u n ∣ q d ξ + λ 1 4 − 1 q ∫ Ω ∣ u n ∣ q ln ∣ u n ∣ 2 d ξ = o ( 1 ) .

This is impossible because c > 0 . Hence A > 0 and so Proposition 2.1, (2.3), (2.16), and the fact that ⟨ J λ ′ ( u n ) , u n ⟩ = o ( 1 ) yield

(2.17) M ( ‖ u n ‖ 2 ) ‖ u n ‖ 2 = − μ ∫ Ω ϕ u ∣ u ∣ 2 d ξ + λ ∫ Ω ∣ u ∣ q ln ∣ u ∣ 2 d ξ + ∫ Ω ∣ u ∣ 4 d ξ + o ( 1 ) .

Since ⟨ J λ ′ ( u n ) , v ⟩ = o ( 1 ) for any v ∈ S 0 1 ( Ω ) , one sees by (2.7) that

(2.18) M ( A ) ⟨ u , v ⟩ = − μ ∫ Ω ϕ u u v d ξ + λ ∫ Ω ∣ u ∣ q − 2 u v ln ∣ u ∣ 2 d ξ + ∫ Ω ∣ u ∣ 2 u v d ξ .

Therefore, (2.17) and (2.18), with v = u , give at once that M ( A ) ‖ u ‖ 2 = M ( ‖ u n ‖ 2 ) ‖ u n ‖ 2 + o ( 1 ) . Hence, ( M 1 ) proves claim (2.8) and completes the proof.□

3 Proof of Theorem 1.1

Now we are in a position to prove Theorem 1.1, and we assume that the hypotheses of Theorem 1.1 are satisfied. We need the mountain pass theorem in the following version.

Proposition 3.1

(see [1]) Let E be a real Banach space and the functional ℐ ∈ C 1 ( E , R ) satisfies ℐ ( 0 ) = 0 and

there are constants ρ , α > 0 such that inf ‖ u ‖ = ρ ℐ ≥ α ;
there is e ∈ E \ B ρ such that ℐ ( e ) < 0 .

Let c be defined by

c = inf γ ∈ Γ max t ∈ [ 0 , 1 ] ℐ ( γ ( t ) ) , with Γ = { γ ∈ C ( [ 0 , 1 ] , E ) : γ ( 0 ) = 0 , ℐ ( γ ( 1 ) ) < 0 } .

If ℐ satisfies the ( P S ) c condition, then c is a critical value for ℐ and c ≥ α .

Lemma 3.1

The functional J λ has a mountain pass geometry in S 0 1 ( Ω ) .

Proof

From ( M 2 ), ( M 3 ), (2.2), and the Sobolev embedding inequality, we have

(3.1) J λ ( u ) ≥ 1 2 θ M ( ‖ u ‖ 2 ) ‖ u ‖ 2 + μ 4 ∫ Ω ϕ u ∣ u ∣ 2 d ξ − λ q ∫ Ω ( ε ∣ u ∣ 2 θ + C ε ∣ u ∣ r ) d ξ − 1 4 ∫ Ω ∣ u ∣ 4 d ξ ≥ 1 2 θ − λ q ε C 1 m 1 ‖ u ‖ 2 θ − λ q C ε C 2 ‖ u ‖ r − C 3 ( ∣ μ ∣ + 1 ) ‖ u ‖ 4 .

Thus, choosing ε = q / 4 λ C 1 θ > 0 and ρ > 0 small enough for all u ∈ S 0 1 ( Ω ) with ‖ u ‖ = ρ , inequality (3.1) gives

J λ ( u ) ≥ 1 4 θ m 1 ρ 2 θ − λ q C ε C 2 ρ r − C 3 ( ∣ μ ∣ + 1 ) ρ 4 ≥ α

for a suitable α > 0 because 2 θ < r . Observe that

(3.2) 2 t q − q t q ln ∣ t ∣ 2 ≤ 2 for all t ∈ R + ,

and for a fixed t 0 > 0 , assumption ( M 2 ) yields that

(3.3) M ^ ( t ) ≤ M ^ ( t 0 ) t 0 θ t θ = C 0 t θ for all t ≥ t 0 .

Take v ∈ S 0 1 ( Ω ) \ { 0 } . Since, by (2.5),

∫ Ω ∣ v ∣ 4 d ξ − μ ∫ Ω ϕ v ∣ v ∣ 2 d ξ > 0 for all μ ∈ − ∞ , S ∣ Ω ∣ − 1 2 ,

we deduce that

J λ ( t v ) ≤ C 0 2 t 2 θ ‖ v ‖ 2 θ + λ q 2 ∣ Ω ∣ − t 4 4 ∫ Ω ∣ v ∣ 4 d ξ − μ ∫ Ω ϕ v ∣ v ∣ 2 d ξ → − ∞ as t → ∞

by (3.3), (3.2), and the fact that θ < 2 . Hence, putting e = t 0 v with t 0 large enough, we obtain J λ ( e ) < 0 . This completes the proof.□

Proof of Theorem 1.1

By Lemmas 2.1 and 3.1 and Proposition 3.1, there exists a nontrivial critical point of J λ at level

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

where Γ λ = { γ ∈ C ( [ 0 , 1 ] , S 0 1 ( Ω ) ) : γ ( 0 ) = 0 , J λ ( γ ( 1 ) ) < 0 } , provided that

(3.4) c λ < 1 2 θ − 1 q m 1 2 2 − θ S 2 θ 2 − θ .

We claim that (3.4) holds true for all λ > 0 large enough.

To prove (3.4), we choose v 0 ∈ S 0 1 ( Ω ) with ‖ v 0 ‖ = 1 . From the proof of Lemma 3.1, it is clear that J λ ( t v 0 ) > 0 for all t > 0 small enough and that J λ ( t v 0 ) → − ∞ as t → ∞ . Hence, there is t λ > 0 such that

J λ ( t λ v 0 ) = sup t ≥ 0 J λ ( t v 0 ) .

Moreover, ⟨ J λ ′ ( t λ v 0 ) , t λ v 0 ⟩ = t λ d d t J λ ( t v 0 ) t = t λ = 0 , that is,

(3.5) M ( t λ 2 ) t λ 2 + μ t λ 4 ∫ Ω ϕ v 0 ∣ v 0 ∣ 2 d ξ = λ ∫ Ω ∣ t λ v 0 ∣ q ln ∣ t λ v 0 ∣ 2 d ξ + t λ 4 ∫ Ω ∣ v 0 ∣ 4 d ξ .

It follows from (3.2), (3.3), and ( M 2 ) that

C 0 θ t λ 2 θ ≥ − 2 λ q ∣ Ω ∣ + t λ 4 ∫ Ω ∣ v 0 ∣ 4 d ξ − μ ∫ Ω ϕ v 0 ∣ v 0 ∣ 2 d ξ .

Since, by (2.5), ∫ Ω ∣ v 0 ∣ 4 d ξ − μ ∫ Ω ϕ v 0 ∣ v 0 ∣ 2 d ξ > 0 for all μ ∈ − ∞ , S ∣ Ω ∣ − 1 2 , the above inequality gives that { t λ } λ > 0 is bounded. Next, we claim that

(3.6) t λ → 0 as λ → ∞ .

Otherwise, there exists a sequence ( λ n ) n with λ n → ∞ such that t λ n → t 0 as n → ∞ for some t 0 > 0 . The dominated convergence theorem gives as n → ∞ ,

∫ Ω ∣ t λ n v 0 ∣ q ln ∣ t λ n v 0 ∣ 2 d ξ → ∫ Ω ∣ t 0 v 0 ∣ q ln ∣ t 0 v 0 ∣ 2 d ξ ,

and so

λ n ∫ Ω ∣ t λ n v 0 ∣ q ln ∣ t λ n v 0 ∣ 2 d ξ → ∞ .

Thanks to (3.5), this contradicts

lim n → ∞ M ( t λ n 2 ) t λ n 2 + μ t λ n 4 ∫ Ω ϕ v 0 ∣ v 0 ∣ 2 d ξ = M ( t 0 2 ) t 0 2 + μ t 0 4 ∫ Ω ϕ v 0 ∣ v 0 ∣ 2 d ξ ( ∈ R )

and proves the latter claim (3.6).

Therefore, using (3.5), (3.6), and the fact that M is continuous at 0, we deduce at once that as λ → ∞ ,

− λ ∫ Ω ∣ t λ v 0 ∣ q ln ∣ t λ v 0 ∣ 2 d ξ = t λ 4 ∫ Ω ∣ v 0 ∣ 4 d ξ − M ( t λ 2 ) t λ 2 − μ t λ 4 ∫ Ω ϕ v 0 ∣ v 0 ∣ 2 d ξ → 0 .

Moreover, also

λ ∫ Ω ∣ t λ v 0 ∣ q d ξ → 0 as λ → ∞ .

By the definition of J λ , it follows that

J λ ( t λ v 0 ) → 0 as λ → ∞ ,

which implies that there exists λ ∗ > 0 such that for λ > λ ∗ ,

c λ ≤ sup t ≥ 0 J λ ( t v 0 ) = J λ ( t λ v 0 ) < 1 2 θ − 1 q m 1 2 2 − θ S 2 θ 2 − θ ,

i.e., claim (3.4) holds true. The proof of Theorem 1.1 is now complete.□

Funding information: P. Pucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Projects and partly supported by GNAMPA–INdAM Project 2022 Equazioni differenziali alle derivate parziali in fenomeni non lineari (CUP_E55F22000270001). P. Pucci was also partly supported by the Fondo Ricerca di Base di Ateneo – Esercizio 2017–2019 of the University of Perugia, named PDEs and Nonlinear Analysis. This article was written while Yiwei Ye was visiting the University of Perugia under the auspices of the China Scholarship Council. Yiwei Ye thanks the University of Perugia for the hospitality.
Conflict of interest: Prof. Pucci is a member of the Editorial Board, but she had no involvement in the final decision.

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Received: 2022-02-08

Revised: 2022-07-07

Accepted: 2022-07-11

Published Online: 2022-08-11

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https://doi.org/10.1515/ans-2022-0018

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Heisenberg group; Kirchhoff-Poisson system; critical growth; logarithmic nonlinearity; concentration-compactness principle

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