On double phase Kirchhoff problems with singular nonlinearity

1 Introduction

In this work, we are concerned with multiple solutions for double phase problems with a nonlocal Kirchhoff term and a singular right-hand side. To be more precise, we study the problem

with Ω ⊂ R N ( N ≥ 2 ) being a bounded domain with Lipschitz boundary ∂ Ω , λ > 0 is the parameter to be specified, and ℒ p , q a denotes the double phase operator given by

Furthermore, we suppose the following conditions:

1. 1. 1 < p < N , p < q < p ∗ , and 0 ≤ a ( ⋅ ) ∈ L ∞ ( Ω ) with p ∗ being the critical Sobolev exponent to p given by

      (1.2) p ∗ = N p N − p ;

   2. 0 < γ < 1 and m : [ 0 , ∞ ) → [ 0 , ∞ ) is a continuous function defined by

      (1.3) m ( t ) = a 0 + b 0 t θ − 1 for all t ≥ 0 ,

      where a 0 ≥ 0 , b 0 > 0 with θ ∈ 1 , r q and r ∈ ( q θ , p ∗ ) .

Problems of type ( P λ ) combine several interesting phenomena into one problem. First, the differential operator involved is the so-called double phase operator given in (1.1). In 1986, Zhikov [37] introduced for the first time in the literature the related energy functional to (1.1) defined by

This kind of functional has been used to describe models for strongly anisotropic materials in the context of homogenization and elasticity. Indeed, the hardening properties of strongly anisotropic materials change point by point. For this, the modulating coefficient a ( ⋅ ) helps to regulate the mixture of two different materials, with hardening powers p and q . From the mathematical point of view, the behavior of (1.4) is related to the sets on which the weight function a ( ⋅ ) vanishes or not. Hence, there are two phases a ( x ) = 0 and a ( x ) ≠ 0 , and so (1.4) is said to be of double phase type. In this direction, functional (1.4) has several mathematical applications in the study of duality theory and of the Lavrentiev gap phenomenon, see Zhikov [38,39]. Also, (1.4) belongs to the class of the integral functionals with nonstandard growth condition, according to Marcellini’s terminology [30,31]. Following this line of research, Mingione et al. provide famous results in the regularity theory of local minimizers of (1.4), see, for example, the works by Baroni et al. [4,5] and Colombo and Mingione [8,9].

A second interesting phenomenon is the appearance of a nonlocal Kirchhoff term given in (1.3), which was first introduced by Kirchhoff [27]. Problems as in ( P λ ) involving a Kirchhoff term are said to be degenerate if a 0 = 0 and nondegenerate if a 0 > 0 . It is worth noting that the degenerate case is rather interesting and is treated in well-known articles in the Kirchhoff theory. We do cover the degenerate case in our paper, which has several applications in physics. For example, the transverse oscillations of a stretched string with nonlocal flexural rigidity depends continuously on the Sobolev deflection norm of u via m ( ∫ Ω ∣ ∇ u ∣ 2 d x ) , that is, m ( 0 ) = 0 is nothing less than the base tension of the string is zero.

A third fascinating aspect of our problem is the presence of a nonlinear singular term in ( P λ ) . The study of elliptic or integral equations involving singular terms started in the early sixties by the work of Fulks and Maybee [18], originating from the models of heat conduction in electrically conducting materials. More precisely, let Ω be an electrically conducting medium in R 3 and u be the steady-state temperature distribution in the region Ω . Then, if λ u γ is the rate of generation of heat with constant voltage λ (as in our model ( P λ ) ), then the temperature distribution in the conducting medium satisfies the local and linear counterpart of the equation mentioned in ( P λ ) . For interested readers, we refer to the works of Diaz et al. [11], Nachman and Callegari [32], and Stuart [34] for applications in non-Newtonian fluid flows in porous media and heterogeneous catalysts, pseudo-plastic fluids, and in the theory of radiative transfer in semi-infinite atmospheres, respectively.

Denoting

and indicating with W 0 1 , ℋ ( Ω ) the homogeneous Musielak-Orlicz Sobolev space, which will be introduced in Section 2, we can state the following definition of a weak solution to problem ( P λ ) .

Based on hypotheses (H) and Proposition 2.1 in Section 2, it is clear that the definition of a weak solution is well defined. Moreover, we introduce the corresponding energy functional J λ : W 0 1 , ℋ ( Ω ) → R associated to problem ( P λ ) defined by

where M : [ 0 , ∞ ) → [ 0 , ∞ ) is given by

The main result in this paper reads as follows.

The proof of Theorem 1.2 is based on a careful study of the corresponding fibering map, which was initiated by the work of Drábek and Pohozaev [12]. The idea is to define the related Nehari manifold of ( P λ ) in the form of the first derivative of the fibering mapping, and then we split the Nehari manifold into three disjoint parts, which are related to the second derivative of the fibering function. It turns out that the global minimizers of J λ restricted to two of them are the solutions we seek and the third one is the empty set for small values of the parameter λ > 0 . This method has become a very powerful tool and has been used in several papers.

To the best of our knowledge, the only work dealing with a double phase operator and a nonlocal Kirchhoff term has been recently done by Fiscella and Pinamonti [17] who studied the problem

with a Carathéodory f : Ω × R → R satisfying subcritical growth and the Ambrosetti-Rabinowitz condition. Based on the mountain-pass theorem, the existence of a nontrivial weak solution of (1.5) is shown. In addition, the authors in [17] considered the problem

and proved the existence of infinitely many weak solutions with unbounded energy by using the fountain theorem. Even if the double phase operator does not explicitly appear in (1.6), this problem has still a variational structure set in the same double phase framework of (1.5). Further results in the context of double phase Kirchhoff problems can be found in the recent works of Arora et al. [1], Fiscella et al. [16], Gupta and Dwivedi [23], Ho and Winkert [25] and Isernia and Repovš [26].

Finally, we mention some existence and multiplicity results for double phase problems without Kirchhoff term, that is, m ( t ) ≡ 1 for all t ≥ 0 . We refer to the papers of Arora and Shmarev [2,3] (parabolic double phase problems), Colasuonno and Squassina [7] (eigenvalue problems), Farkas and Winkert [14], Farkas et al. [13] (singular Finsler double phase problems), Fiscella [15] (Hardy potentials), Gasiński and Papageorgiou [19] (locally Lipschitz right-hand side), Gasiński and Winkert [20–22] (convection and superlinear problems), Liu and Dai [28] (Nehari manifold approach), Liu et al. [29] (singular problems), Perera and Squassina [33] (Morse theoretical approach), Zeng et al. [35,36] (multivalued obstacle problems), and the references therein.

This paper is organized as follows. In Section 2, we recall the main properties of Musielak-Orlicz Sobolev spaces W 0 1 , ℋ ( Ω ) and state the main embeddings concerning these spaces. Section 3 gives a detailed analysis of the fibering map and presents the main properties of the three disjoints subsets of the Nehari manifold. In Section 4, we prove the existence of at least two weak solutions of problem ( P λ ) , see Propositions 4.4 and 4.6. Finally, in Section 5, we study a singular Kirchhoff problem driven by the left-hand side of (1.6), inspired by [17].

2 Preliminaries

In this section, we will recall the main properties and embedding results for Musielak-Orlicz Sobolev spaces. To this end, we suppose that Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with Lipschitz boundary ∂ Ω . For any r ∈ [ 1 , ∞ ) , we denote by L r ( Ω ) = L r ( Ω ; R ) and L r ( Ω ; R N ) the usual Lebesgue spaces with the norm ‖ ⋅ ‖ r . Moreover, the Sobolev space W 0 1 , r ( Ω ) is equipped with the equivalent norm ‖ ∇ ⋅ ‖ r for 1 < r < ∞ .

Let hypothesis (H)(i) be satisfied and consider the nonlinear function ℋ : Ω × [ 0 , ∞ ) → [ 0 , ∞ ) defined by

Denoting by M ( Ω ) the space of all measurable functions u : Ω → R , we can introduce the Musielak-Orlicz Lebesgue space L ℋ ( Ω ) , which is given by

equipped with the Luxemburg norm

where the modular function is given by

The norm ‖ ⋅ ‖ ℋ and the modular function ϱ ℋ have the following relations, see Liu and Dai [28, Proposition 2.1] or Crespo-Blanco et al. [10, Proposition 2.13].

Furthermore, we define the seminormed space

endowed with the seminorm

While, the corresponding Musielak-Orlicz Sobolev space W 1 , ℋ ( Ω ) is defined by

equipped with the norm

where ‖ ∇ u ‖ ℋ = ‖ ∣ ∇ u ∣ ‖ ℋ . Moreover, we denote by W 0 1 , ℋ ( Ω ) the completion of C 0 ∞ ( Ω ) in W 1 , ℋ ( Ω ) . From hypothesis (H)(i), we know that we can equip the space W 0 1 , ℋ ( Ω ) with the equivalent norm given by

see Proposition 2.16(ii) of Crespo-Blanco et al. [10]. It is known that L ℋ ( Ω ) , W 1 , ℋ ( Ω ) and W 0 1 , ℋ ( Ω ) are uniformly convex and so reflexive Banach spaces, see Colasuonno-Squassina [7, Proposition 2.14] or Harjulehto-Hästö [24, Theorem 6.1.4].

We end this section by recalling the following embeddings for the spaces L ℋ ( Ω ) and W 0 1 , ℋ ( Ω ) , see Colasuonno and Squassina [7, Proposition 2.15] or Crespo-Blanco et al. [10, Propositions 2.17 and 2.19].

3 Analysis of the fibering function

As mentioned in Section 1, the proof of Theorem 1.2 relies on the fibering map corresponding to our problem. For this purpose, we recall that the energy functional J λ : W 0 1 , ℋ ( Ω ) → R related to problem ( P λ ) is given by

Due to the presence of the singular term, we know that J λ is not C 1 . For u ∈ W 0 1 , ℋ ( Ω ) \ { 0 } , we introduce the fibering function ψ u : [ 0 , ∞ ) → R defined by

which gives

Note that ψ u ∈ C ∞ ( ( 0 , ∞ ) ) . In particular, we have for t > 0

and

On this basis, we can introduce the Nehari manifold related to our problem, which is defined by

Therefore, we have u ∈ N λ if and only if

Also t u ∈ N λ if and only if ψ ′ ( 1 ) = 0 . It is clear that N λ contains all weak solutions of ( P λ ) but it is smaller than the whole space W 0 1 , ℋ ( Ω ) . For our further study, we need to decompose the set N λ in the following disjoints sets:

Now, we show the coercivity of the energy functional J λ restricted to the Nehari manifold N λ .

Let S be the best Sobolev constant in W 0 1 , p ( Ω ) defined as follows:

The next result shows the emptiness of N λ ∘ for small values of λ .

Let us now analyze the map ψ u ′ ( t ) in more detail. First, we can write