Kirchhoff–Hardy Fractional Problems with Lack of Compactness

1 Introduction

Recently, great attention has been drawn to the study of fractional and nonlocal elliptic problems with lack of compactness. These models arise in a quite natural way in many different applications, and we refer to the recent monograph [27], to the extensive paper [10] and the references cited therein for further details.

This paper is devoted to the study of the existence of solutions of a series of Dirichlet problems in general open subsets Ω of ℝN, N≥1, possibly unbounded. The problems involve a Hardy coefficient and the so-called fractional p-Laplace operator (-Δ)ps, with 0<s<1<p<∞ and p⁢s<N. Hence, ps*=p⁢N/(N-p⁢s) is well defined and is critical in the sense of the fractional Sobolev theory. The operator (-Δ)ps (up to normalization factors) is defined for any x∈ℝN by

along any φ∈C0∞⁢(ℝN), where Bε⁢(x) is the open ball of ℝN centered at x and with radius ε>0.

Let 0≤α<p⁢s and let ps*⁢(α)=p⁢(N-α)/(N-p⁢s)≤ps*⁢(0)=ps*. The main results of the paper are based on the best fractional Hardy–Sobolev constant Hα=H⁢(p,N,s,α), given by

where Z⁢(Ω) is the completion of C0∞⁢(Ω), with respect to the norm

well defined along any test function φ∈C0∞⁢(Ω), extended to the entire ℝN by putting φ=0 in ℝN∖Ω. We refer to Section 2 for details.

The constant Hα is well defined and strictly positive thanks to Lemma 2.1. However, the fractional Hardy embedding Z⁢(Ω)↪Lps*⁢(α)⁢(Ω,|x|-α) is continuous but not compact. In order to handle the critical Hardy–Sobolev potential as well as the nonlocal term given by (-Δ)ps, we first study the exact behavior of weakly convergent sequences of Z⁢(Ω) in the space of measures. This behavior is described in the following theorem, where the assumption that Ω is bounded seems to play an essential role. If Ω is a bounded open subset of ℝN, then (1.1) reduces simply to the Poincaré theorem when α=p⁢s, that is, (1.1) holds for all α∈[0,p⁢s], in other words for all ps*⁢(α)∈[p,ps*].

For the case α=0 we refer to [29, Theorem 2.5]. As a direct consequence of Theorem 1.1 above and [29, Theorem 2.5], we prove that the functional

is weakly lower semi-continuous and coercive in Z⁢(Ω), provided that α, θ, γ, and λ verify suitable restrictions, depending on the behavior of the Kirchhoff coefficient M, which is assumed to satisfy condition

1. M : ℝ 0 + → ℝ 0 + is continuous and nondecreasing. There exist numbers c>0 and θ such that for all t∈ℝ0+,

   (1.7) ℳ ⁢ ( t ) ≥ c ⁢ t θ , with  ⁢ θ ⁢ { ∈ ( 1 , p s * ⁢ ( α ) / p ) ⁢  and  ⁢ α ∈ [ 0 , p ⁢ s )  if  ⁢ M ⁢ ( 0 ) = 0 , = 1 ⁢  and  ⁢ α ∈ [ 0 , p ⁢ s ]  if  ⁢ M ⁢ ( 0 ) > 0 .

A typical prototype for M, due to Kirchhoff, satisfying (ℳ) is given by

with c=a and θ=1 if M⁢(0)>0, while c=b and θ=ϑ if M⁢(0)=0, that is, a=0. Indeed, if M⁢(0)=0 for (1.8), then ϑ>1 as a corollary of [7, Lemma 3.1].

The functional ℋγ,λ is the basis of the elliptic part of some nonlinear Kirchhoff problems which are studied in Section 3. Theorem 1.1 is also applied in minimization arguments and critical point theorems to get existence and multiplicity results.

In general, when M⁢(t)>0 for all t∈ℝ+, then the related Kirchhoff problem is said to be non-degenerate when M⁢(0)>0, while it is called degenerate if M⁢(0)=0.

In the second part of the paper, we treat Kirchhoff problems in general open sets Ω, with possibly Ω=ℝN. The first problem is

where 0<s<1<p<∞ and 0≤α<p⁢s<N, while σ is a real parameter. Naturally, the condition u=0 in ℝN∖Ω disappears when Ω=ℝN. The exponent q satisfies p⁢θ<q<ps*⁢(α)≤ps*. The norm [⋅]s,p is defined in (1.2).

Since 0≤α<p⁢s<N in problem (1.9), we assume that the nonlocal Kirchhoff term satisfies assumption

1. M : ℝ 0 + → ℝ 0 + is a continuous function such that the following conditions hold:

   1. there exists θ∈[1,ps*⁢(α)/p) such that t⁢M⁢(t)≤θ⁢ℳ⁢(t) for any t∈ℝ0+, where ℳ is defined in (1.6);

   and either

   1. inf t ∈ ℝ 0 + ⁡ M ⁢ ( t ) = a > 0 ;

   or M⁢(0)=0 and M satisfies both properties

   1. for any τ>0 there exists m=m⁢(τ)>0 such that M⁢(t)≥m for all t≥τ;

   2. there exists a positive number c>0 such that M⁢(t)≥c⁢tθ-1 for all t∈[0,1].

Clearly, condition (M2~) covers the so-called non-degenerate case and implies at once the validity of (M2) and (M3). Concerning the positive weight w, we assume

1. w ∈ L ℘ ⁢ ( ℝ N ) , with ℘=ps*/(ps*-q) and 1<q<ps*.

Condition (w) guarantees that the embedding Z⁢(Ω)↪Lq⁢(Ω,w) is compact, even when Ω is the entire ℝN, as explained in Section 4. Indeed, the natural solution space for problem (1.9) is the fractional density space Z⁢(Ω), that is, the closure of C0∞⁢(Ω) with respect to [⋅]s,p, given in (1.2). Thus,

with Cw=H0-1/p⁢∥w∥℘1/q>0, as proved in Lemma 4.1.

In the next result, we partially answer the open question asked for the Kirchhoff–Hardy equation [7, (1.1)] since we cover for a slightly different equation also the degenerate case. Thanks to the variational nature of (1.9), (weak) solutions of (1.9) are exactly the critical points of the underlying functional 𝒥σ, which satisfies the geometry of the mountain pass lemma under the above structural assumptions. The critical points uσ of 𝒥σ in Z⁢(Ω) are found at special mountain pass levels cσ, and these solutions of (1.9) are simply called mountain pass solutions.

The degenerate nature of problem (1.9) does not allow us to apply Theorem 1.1 in the proof of Theorem 1.2. As is customary in elliptic problems, involving critical Hardy nonlinearities, the delicate point is the verification of the Palais–Smale condition. For this we exploit an asymptotic property of the mountain pass level cσ, taking inspiration from the proof of [15, Theorem 1.3] and also for a somehow similar problem from the proof of [1, Theorem 1.1].

The case Ω=ℝN and α=0 of Theorem 1.2 was first treated in [7, Theorem 1.2]. Furthermore, Theorem 1.2 extends in several directions [1, Theorem 1.1 and Theorem 1.2 (i)], [12, Theorem 1.1], [22, Theorem 1.1], [24, Theorem 1.1], [26, Theorem 1.1], and [30, Theorem 1.1 (ii)].

Moreover, in the non-degenerate case, following [1, Theorem 1.2 (ii)], we have this nice addition.

As explained in the introduction of [1], the request (1.12) is automatic whenever M⁢(0)=a, with p<q. The case M⁢(0)=a occurs in the non-degenerate prototype case (1.8), and more generally, whenever M is monotone increasing in ℝ0+. As we shall see, Theorem 1.3 is proved via a truncation argument on M since the Kirchhoff function M could increase too quickly with respect to the other terms of problem (1.9). Theorem 1.3 extends in several directions [1, Theorem 1.2 (ii)], in particular to the case in which Ω could be possibly unbounded and also to the case Ω=ℝN. Furthermore, Theorem 1.3 generalizes, e.g., [24, Theorem 1.2 (2) and (3)] and [12, Theorem 1.1].

In the last part of the work, we study the nonhomogeneous version of the Kirchhoff problem (1.9), considering M of the special type (1.8), with θ replacing ϑ for simplicity. Actually, we treat the general problem

with α∈[0,p⁢s) but β∈[0,p⁢s). Of course, the only interesting case occurs when θ>1, and we assume it without loss of generality, with possibly b=0. Hence, the addition of a sufficiently small nontrivial perturbation allows us to balance both Hardy–Sobolev terms with the Kirchhoff coefficient M, possibly zero at zero. The latter Hardy–Sobolev term could coincide with the Sobolev critical nonlinearity when β=0, a special interesting case.

The result of Theorem 1.4 can be summarized in Table 1.

The values δ>0 and σ*∈(0,∞] are constructed in the technical Lemma 4.5. It is worth noting that p⁢θ<ps* since p⁢θ<ps*⁢(β)≤ps*. Hence, p⁢θ cannot be equal to ps*⁢(α)=ps*⁢(0)=ps* when α=0.

The strategy used in the proof of Theorem 1.2 seems not to work for (1.13). This is why we had to require that M is of the special canonical form since this allows us to prove that the weak limit of a minimizing sequence is actually a nontrivial local interior minimum point of the functional corresponding to (1.13), that is, a nontrivial solution of (1.13).

Theorem 1.4 extends in several directions [24, Theorems 1.3 and 1.4], and generalizes the existence part contained in the multiplicity results given in [10, Theorem 2.1.1], [14, Theorem 1.1], [21, Theorem 1.1], [26, Theorem 1.2], and [35, Theorem 1.1].

A natural appealing open problem is to prove the existence of nontrivial solutions for (1.9) and (1.13) when α=p⁢s, a case not covered in Theorems 1.2–1.4. When α=p⁢s, β=0 and g≡0, a problem somehow related to (1.13) is treated in details in [7, Theorem 1.1], but only in the non-degenerate setting, that is, under condition (M2~) and under a suitable geometrical restriction on γ. Finally, the case α=p⁢s and β=0 is covered in [32, Theorem 1.3] for a problem similar to (1.13), but again in the non-degenerate setting, that is, when a>0.

The paper is organized as follows: In Section 2, we prove Theorem 1.1 with a result from Appendix A. In Section 3, we provide some applications of Theorem 1.1. Section 4 contains the proofs of Theorems 1.2–1.4. Finally, in Section 5, we extend the previous results when (-Δ)ps is replaced by a nonlocal integro-differential operator ℒKp, generated by a general singular kernel K, satisfying the natural assumptions described by Caffarelli, e.g., in [6]; see also [33, 34].

2 A Concentration-Compactness Result

This section is devoted to the proof of Theorem 1.1, which concerns the delicate study of the exact behavior of weakly convergent sequences of Z⁢(Ω) in the space of measures.

Let us first introduce the fractional Hardy–Sobolev inequality which is basic for (1.1). By [25, Theorems 1 and 2], we know that

for all u∈Ds,p⁢(ℝN), where cN,p is a positive constant depending only on N and p and Ds,p⁢(ℝN) is the fractional Beppo–Levi space, that is, the completion of C0∞⁢(ℝN), with respect to the norm [⋅]s,p defined in (1.2).

Let Ω be any open set of ℝN and let Z⁢(Ω) be the completion of C0∞⁢(Ω), with respect to the norm [⋅]s,p defined in (1.2). When Ω is bounded, p=2 and K⁢(x)=|x|-N-2⁢s, then Z⁢(Ω) is equivalent to the Hilbert space defined in [13]. Even if Z⁢(Ω) is not a real space of functions, but a density space, the choice of this solution space is an improvement with respect to the space

fairly popular in recent papers devoted to nonlocal variational problems. Indeed, the density result proved in [16, Theorem 6] does not hold true for X0⁢(Ω) without assuming more restrictive conditions on the open bounded set Ω and on its boundary ∂⁡Ω; see in particular [16, Remark 7]. In conclusion, if Ω is an open bounded subset of ℝN, then Z⁢(Ω)⊂X0⁢(Ω), with possibly Z⁢(Ω)≠X0⁢(Ω).

It is worth noting that if Ω is any open subset of ℝN and u~ denotes the natural extension of any u∈Z⁢(Ω), then u~∈Ds,p⁢(ℝN) by (2.1). In other words,

and equality holds when either Ω=ℝN or ∂⁡Ω is continuous by [20, Theorem 1.4.2.2]. In what follows, with abuse of notation, we continue to write u in place of u~ since the context is clear.

Therefore, the main density function space Z⁢(ℝN) reduces to Ds,p⁢(ℝN), and so

Clearly, Z⁢(ℝN)=Ds,p⁢(ℝN) is the suitable solution space for problems (1.9) and (1.13) when Ω=ℝN.

Thus, by the interpolation and the Hölder inequalities we easily get the next fractional Hardy–Sobolev inequality, proved for p=2 in [19, Lemma 2.1]. However, for the sake of completeness we give the proof when 0<α<p⁢s<N since when either α=0 or α=p⁢s, the fractional Hardy–Sobolev inequality reduces exactly to (2.1).

From Lemma 2.1 it is clear that the fractional Sobolev embedding Z⁢(ℝN)↪Lps*⁢(ℝN) and the fractional Hardy–Sobolev embedding Z⁢(ℝN)↪Lps*⁢(α)⁢(ℝN,|x|-α) are continuous, but not compact. However, we are able to introduce the best fractional Hardy–Sobolev constant Hα=H⁢(p,N,s,α), as stated in (1.1). Of course, the number Hα is strictly positive and it coincides with the best fractional Sobolev constant when α=0. Consequently, the embeddings Z⁢(Ω)↪Lps*⁢(Ω) and Z⁢(Ω)↪Lps*⁢(α)⁢(Ω,|x|-α) are continuous, but not compact.

The last part of this section is devoted to the proof of Theorem 1.1. Even if the proof of Theorem 1.1 is fairly similar to that of [15, Theorem 2.2] given in the case p=2, here we do not use any longer [31, Lemmas 5 and 6], where the requirement that Ω is bounded seems to be crucial. However, the proof of Theorem 1.1 is based on the tightness of the sequence (|Ds⁢uj|p)j, and the tightness property is obtained as an application of [3, Theorem 8.6.2]. It is exactly at this step that we use that Ω is bounded.