On Critical Kirchhoff problems driven by the fractional Laplacian

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the sense of Sobolev embeddings. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.


Introduction
The equation that goes under the name of Kirchho equation was proposed in [16] as a model for the transverse oscillation of a stretched string in the form We refer to [24] for a recent survey of the results connected to this model.
The existence and multiplicity of solutions to Kirchho problems under the e ect of a critical nonlinearity have received considerable attention. The term critical refers here to the rough assumption that ( ) ∼ | | 2 * −2 with 2 * = 2 /( − 2). The natural setting of the corresponding equation in 1 0 ( ) yields a lack of compactness, since the embedding of 1 0 ( ) into 2 * ( ) is only continuous. Straightforward techniques of Calculus of Variations thus fail, and more advanced results from Critical Point Theory must be used. In particular, P.-L. Lions' Concentration-Compactness appears as a natural tool for the analysis of the loss of compactness. The relevant outcome is that the Kirchho function interacts with the critical growth of the nonlinearity : the validity of the Palais-Smale compactness condition holds only under a condition like −4 2 ≥ 2 ( ), and a similar inequality ensures that the associated Euler functional is weakly lower semicontinuous.
In the recent paper [13], Faraci  where is an open bounded subset of ℝ , > 4, and are positive xed numbers, is a real parameter and is a Carathéodory function that satis es suitable growth conditions. By using a bering-type approach, the authors of [13] investigate existence, non-existence and multiplicity of solutions to (1.4). In the previous paper [14], Faraci, Farkas and Kristály studied equation (1.4) with ( , ) = 0 and under suitable assumptions on the parameters and they proved that the functional associated to the problem is sequentially weakly lower semicontinuous, satis es the Palais-Smale condition and is convex.
The purpose of the present paper is to extend part of these results to the fractional counterpart of the Kirchho problem where ⊂ ℝ is a bounded domain with Lipschitz boundary ,  = ℝ 2 ⧵  and  = × , and are strictly positive real numbers, ∈ (0, 1), > 4 and 2 * ∶= 2 /( − 2 ) denotes the critical exponent for the Sobolev embedding of (ℝ ) into Lebesgue spaces. is a function that satis es hypothesis similar to the one in (1.4) adapted to the non local case. The fractional Laplacian in ( , ) is de ned as Since the parameter is xed, we will work with a rescaled version of the operator and this enables us to assume that , = 1. For references about the fractional Laplacian we refer to [11], [1] and to the monograph [21]. We de ne the space as the set of functions ∶ ℝ → ℝ such that | ∈ 2 ( ) and (1.5) We also set 0 ( ) ∶= ∈ ∶ = 0 a.e. in ℝ ⧵ .
We introduce the best Sobolev constant for the continuous embedding 0 ( ) ⊂ 2 * ( ) as This norm is induced by the scalar product for all , ∈ 0 ( ) and we recall that in 0 ( ) it is equivalent to (1.5). For further details we refer the reader to [25,Lemma 6]. Weak solutions to ( , ) correspond to critical points of the functional  , ∶ 0 ( ) → ℝ associated to the problem: where we denote with ( , ) = 0 ( , ) . Arguing as in [27,Proposition 1.12], we get for all , ∈ 0 ( ). When we have ( , ) = 0 we will use the notation and we point out that  , is a 2 -functional. The interest in generalizing to the fractional case the model introduced by Kirchho does not arise only for mathematical purposes. In fact, following the ideas of [6] and the concept of fractional perimeter, Fiscella and Valdinoci proposed in [15] an equation describing the behavior of a string constrained at the extrema in which appears the fractional length of the rope. The interested reader can also consult [7,8,9] and the references therein for further motivations and applications of operators similar to the one proposed in ( , ).
Recently, problems similar to (1.5) have been extensively investigated by many authors using di erent techniques and producing several relevant results. In [15] Fiscella and Valdinoci showed the existence of a non-negative solution of mountain pass type for an equation with a critical term perturbed with a subcritical nonlinearity. In the same spirit, Autuori, Fiscella and Pucci generalize in [4] these results to the degenerate case, i.e (0) = 0, without any monotonicity assumption on the function . Iin these two articles the operator taken into account is more general than the one we consider here, but the two coincide making a particular choice on the kernel; see also [22]. Liu, Squassina and Zhang studied in [19] ground state solutions for the Kirchho equation plus a potential with a non linear term asymptotic to a power with critical growth in low dimension. It is also worth mentioning [20] where Mingqi, Rǎdulescu and Zhang proved the existence of nontrivial radial solutions in the non-degenerate and degenerate cases for the non local Kirchho problem in which the fractional Laplacian is replaced by the fractional magnetic operator.
On the other hand, we could not nd any overview of di erent kind of solutions at di erent level of energy for the fractional Kirchho problem. Although some of the results we are going to prove are known, we present a proof based on a adaptation to the fractional case due to Palatucci and Pisante ( [23]) of the Lions second concentration-compactness principle; for the original version of the lemma we refer to [18], as well as [17].
We collect here our main results. Theorem 1.1 guarantees the validity of some crucial properties such as the sequentially weakly lower semicontinuity and the Palais-Smale condition. As we are going to see in the next statement, these facts enable us to use traditional variational methods to completely describe the situation for problem ( , ). We start with two results about the existence of global minimizers at di erent level of energy. There exists 0 ≥ 0 such that for any > 0 it is possible to nd ∈ 0 ( ) ⧵ {0} such that  , ( ) = < 0. In the next Theorem we give some information on what happens when we do not keep xed the parameters , and . It asserts we have some kind of stability when the product ( −4 )/2 becomes close to , . Next statement shows the existence of solution of mountain pass type when ≥ 0 . and Finally we focus on the case ∈ ( 0 − , 0 ), for some small > 0.
Theorem 1.6. There exist > 0, > 0 such that for any ∈ ( 0 − , 0 ) the valuê and Our paper is organized as follows: in Section 1 we present the classic Kirchho model, its generalization to the non local case and we collect in a synthetic way our main results. In Section 2 we prove for the functional associated to the problem with ( , ) = 0 the weak lower semicontinuity, the validity of the Palais-Smale condition and the convexity under suitable assumption on the parameters and . Since the perturbation will have a subcritical growth, we prove these conditions for the problem with the pure power in order to ease notation. The general case requires only minor adjustments. In Section 3 we prove the existence of global minimizers, local minimizers and mountain pass type solutions with di erent energy level at varying of the parameter . At the end of Section 3, strengthening the hypothesis on the non linear term , we are able to give also a non existence result for problem ( , ).

Semicontinuity and the validity of the Palais-Smale condition
In this section we completely describe the range of parameters and for which the functional  , associated to the problem is (sequentially) weakly lower semicontinuous.
Since 2 * < 4 we have that  , is coercive, and from that we can deduce the boundedness of the sequence { } . From [26,Lemma 9], up to a subsequence, we have Using the Hölder inequality, it is straightforward to see that the sequence { } is also bounded in the space ( ), thus there exists two nite measures and such that Since the sequence { } is still bounded in 0 ( ), we have that as → ∞. By using the Hölder inequality, we estimate the rst term of (2.13) for some > 0. As in [3, Lemma 2.1], we have that (2.14) Regarding the second term of (2.13), recalling (2.11), we get We de nẽ
Using a variational approach, we investigate the existence of critical points of the functional de ned on the space 0 ( ) where we denote with ( , ) = 0 ( , ) .
We begin the treatment of our problem by proving a series of technical results that will be useful throughout this section. ( ) for every > 0 it holds Proof. From the boundedness of it follows as in [10] that where in the last expression we used the Sobolev inequality. Dividing by 2 we get the rst statement. ( ) follows similarly.
As we did in the previous section, we show in the following lemma that the functional  , is sequentially lower semicontinuous and satis es the Palais-Smale condition for and su ciently large. Using the Sobolev inequality, taking appropriately and choosing even smaller if necessary we obtain the rst part of the statement. In order to complete the proof, it is su cient to remember that has subcritical growth and to notice that 2 < < 2 * < 4.
Now we choose ∈ 0 ( ) and we consider the system in the unknowns and .

Proposition 3.6. Let and be two topological space, and assume that is compact. Let ℎ ∶ × → ℝ be a continuous function. Then the functionĥ( ) ∶= inf ∈ ℎ( , ) is continuous on .
Proof. We rst observe that for any ∈ the functionĥ is well de ned since is compact and the in mum is always attained at some point ( ) ∈ . Recalling that the sets (−∞, ) and ( , ∞) for some , ∈ ℝ form a subbase of ℝ, our proof is reduced to the following: Thus, we can conclude observing that Remark 3.7. We can even strengthen the result above for functions de ned on non compact spaces requiring divergence at in nity. For instance, suppose ℎ ∶ (ℝ + ) 2 → ℝ is continuous and such that lim →∞ ℎ( , ) = ∞ for any ∈ ℝ + . The proof for sets as (−∞, ) is the same. As regard sets of the type ( , ∞) we observe thatĥ −1 ( , ∞) can be written as in (3.3) plus an half line due to the divergence of the function at in nity.
Proof. De ne the continuous function ℎ( , ) ∶=  , , ( ). We start pointing out that ℎ(0, ) is positive on (0, ∞) (see Remark 3.1) and goes to +∞ as → ∞. By continuity we have that for small ℎ( , ) is nonnegative for all ∈ ℝ + . Moreover, from Proposition 3.4 it follows that for any ≥ 0 there is a neighbourhood such that ℎ( , ) > 0 for all ∈ ∩ (0, ∞). We also have that ℎ( , ) → −∞ as → ∞ for any > 0. At this point we de ne the continuous function (refer to Proposition 3.6) ( ) = inf ∈[0,∞) ℎ( , ). From the previous considerations, we can deduce that for su ciently large the function is negative, while if we restrict it is equal to zero. This is due to the fact that the function ℎ( , ) for big enough has a global minimizer in the variable at a negative level. Shrinking , and remembering that all continuous functions are homotopically equivalent, this minimizer becomes local and attained at a positive level. All these arguments ensure us the existence of the desired 0 ( ) that solves (3.2). In addition, Proof. The statement follows immediately from the proof of Proposition 3.8.
Now we de ne a suitable parameter independent from that will play a crucial rôle in the sequel. More precisely, we set The next Proposition shows how the parameter 0 varies depending on the choice made on and . ( ) we have that also ( , 0 ) is a solution of (3.2). From the uniqueness of the parameter 0 ( ) it follows that 0 ( ) = 0 ( ). Now, assume by contradiction that 0 = 0. If that, there is a sequence ( ) ⊂ 0 ( ) ⧵ {0} such that ∶= 0 ( ) → 0. By homogeneity, we may assume that ‖ ‖ = 1. From Proposition 3.8 it follows that there exists > 0 such that  , , ( ) = 0, that is  Hence, 0 ( ) < . We can now let → 0 and we get 0 = 0 as desired. In order to see the last part, let ( ) ⊂ 0 ( ) ⧵ {0} be a sequence such that ∶= 0 ( ) → 0 = 0. As we did in part ), we suppose ‖ ‖ = 1, ⇀ and that there exists > 0 such that which cannot happen since is bounded, see [10].
Next proposition summarize the situation of the in mum depending on the choice of the parameter for the functional  , , ( ).
After some preliminary results we are ready to study the set of solutions of problem ( , ). The rst step will consists in giving the proof for Theorems 1.2 and 1.3 providing the existence of global minimizers for ≥ 0 .
Proof of Theorem 1.2. By the use of assumptions ( 3 ) and ( 4 ) it is easy to verify that  , is coercive. Furthermore, from 3.3 we also have the lower semicontinuity. At this point, as a consequence of the well known Weiestrass Theorem, we have that the in mum is attained. To conclude, we recall that Proposition 3.11 implies the existence of a function in which the functional turns out to be negative. On the other hand , Proposition 3.11 states that  0 , ( ) ≥ 0 for any ∈ 0 ( ), and so 0 =  0 , ( ) = 0. It remains only to prove that is a non trivial minimizer. To see that, observe that Finally, we prove Theorem 1.7 that ensure the existence of mountain pass solutions for < 0 close enough to 0 . In the following we will denote with > 0 the number obtained in Theorem 1.6.