Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity

This paper is concernedwith the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: ⎧ ⎨

where (− ) s N /s is the fractional N /s-Laplacian operator, N ≥ 1, s ∈ (0, 1), ⊂ R N is a bounded domain with Lipschitz boundary, M : R + 0 → R + 0 is a continuous function, and f : × R → R is a continuous function behaving like exp(αt 2 ) as t → ∞ for some α > 0. We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger-Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland's variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger-Moser nonlinearity.

Introduction and main results
In this paper, we study the following fractional Kirchhoff-type problem:  2N dy, x ∈ R N , along functions ϕ ∈ C ∞ 0 (R N ). Throughout this paper, B ε (x) denotes the ball in R N centered at x ∈ R N with radius ε > 0.
To study the existence of solutions for problem (1.1), let us recall some results related to the fractional Sobolev space W s, p 0 ( ). Let 1 < p < ∞ and set 0 ( ) is a uniformly convex Banach space, and hence reflexive, see [38] for more details. The fractional critical exponent is defined by Moreover, the fractional Sobolev embedding theorems states that W s, p For more detailed account on the properties of W s, p 0 ( ), we refer to [10]. In recent years, great attention has been paid to study problems involving fractional operators. In particular, many works focus on the subcritical and critical growth of the nonlinearity which allows us to treat the problem variationally using general critical point theory. Problems like (1.1) are important in many fields of science, notably continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and anomalous diffusion, as they are the typical outcome of stochastically stabilization of Lévy processes, see [1,4,21] and the references therein. Moreover, such equations and the associated fractional operators allow us to develop a generalization of quantum mechanics and also to describe the motion of a chain or an array of particles that are connected by elastic springs as well as unusual diffusion processes in turbulent fluid motions and material transports in fractured media, for more details see [4,5] and the references therein.
Recently, some authors have paid considerable attention in the limiting case of the fractional Sobolev embedding, commonly known as the Trudinger-Moser case. For example, when n = 2, W 1,2 ( ) → L r ( ) for 1 ≤ r < ∞ but W 1,2 ( ) → L ∞ ( ). To fill this gap, Trudinger [37] proved that that there exists τ > 0 such that W 1,2 0 ( ) is embedded into the Orlicz space L φ τ ( ) determined by the Young function φ τ = exp(τ t 2 −1). After that, Moser [25] found the best exponent τ and in particular he obtained a result which is now referred as Trudinger-Moser inequality. In [24], Martinazzi proved that for each u ∈ W s,N /s 0 ( ) and α > 0, there holds exp α|u| Moreover, there exist positive constants were ω N −1 be the surface area of the unit sphere in R N and C N ,s depending only on N and s such that For more details about Trudinger-Moser inequality, we also refer to [19,30]. When N = 1 and s = 1/2, it is still an open problem whether a * N ,s = α N ,s or not. However, for N = 1 and s = 1/2, one can calculate that α N ,s = α * N ,s = 2π 2 and there exists C > 0 such that for all α ∈ [0, 2π 2 ] and the supremum in (1.3) is ∞ for α > 2π 2 .
In the setting of the fractional Laplacian, Iannizzotto and Squassina [17] investigated existence of solutions for the following Dirichlet problem where f (u) behaves like exp(α|u| 2 ) as u → ∞. Using the mountain pass theorem, they obtained the existence of solutions for problem (1.4). Subsequently, Giacomoni, Mishra and Sreenadh [16] studied the multiplicity of solutions for problems like (1.4) by using the Nehari manifold method. Very recently, Perera and Squassina [32] studied the bifurcation results for the following problem with Trudinger-Moser nonlinearity where λ > 0 is a parameter.
For unbounded domains and the general fractional p-Laplacian, Souza [12] considered the following nonhomogeneous fractional p-Laplacian equation where (− ) s p is the fractional p-Laplacian and the nonlinear term f satisfies exponential growth. The author obtained a nontrivial weak solution of the Eq. (1.5) by using fixed point theory. Li and Yang [22] studied the following equation p is the fractional p-Laplacian and f satisfies exponential growth. On the other hand, Li and Yang [23] studied the following Schrödinger-Kirchhoff type equation and f satisfies exponential growth. By using the mountain pass theorem and Ekeland's variational principle, the authors obtained two nontrivial solutions of (1.6) as the parameter λ small enough. Actually, the study of Kirchhoff-type problems, which arise in various models of physical and biological systems, have received more and more attention in recent years. More precisely, Kirchhoff [18] established a model governed by the equation for all x ∈ (0, L), t ≥ 0, where u = u(x, t) is the lateral displacement at the coordinate x and the time t, E is the Young modulus, ρ is the mass density, h is the cross-section area, L is the length and p 0 is the initial axial tension. Equation (1.7) extends the classical D'Alembert wave equation by considering the effects of the changes in the length of the strings during the vibrations. Recently, Fiscella and Valdinoci [14] proposed a stationary Kirchhoff model driven by the fractional Laplacian by taking into account the nonlocal aspect of the tension, see [14, Appendix A] for more details.
In [13], Figueiredo and Severo studied problem (1.8) with N = 2, and the existence of ground state solution obtained by using minimax techniques with the Trudinger-Moser inequality.
Inspired by the above works, especially by [15,28], we are devoted to the existence of ground state solution of (1.1) and overcome the lack of compactness due to the presence of exponential growth terms as well as the degenerate nature of the Kirchhoff coefficient. To the best of our knowledge, there are no results for (1.1) in such a generality.
Throughout the paper, without explicit mention, we assume that M : R + 0 → R + 0 is a continuous function with M(0) = 0, and verifies (M 1 ) for any d > 0 there exists κ := κ(d) > 0 such that M(t) ≥ κ for all t ≥ d; thanks to assumption (M 2 ). Thus, θ M (t) − M(t)t is nondecreasing for t > 0. In particular, we have A typical example of M is given by M(t) = a 0 + b 0 t θ −1 for all t ≥ 0 and some θ > 1, where a 0 , b 0 ≥ 0 and a 0 + b 0 > 0. When M is of this type, problem (1.1) is said to be degenerate if a = 0, while it is called non-degenerate if a > 0. Recently, the fractional Kirchhoff problems have received more and more attention. Some new existence results of solutions for fractional non-degenerate Kirchhoff problems are given, for example, in [33,34,38]. On some recent results concerning about the degenerate case of Kirchhofftype problems, we refer to [3,6,7,26,27,35,[39][40][41] and the references therein. It is worth mentioning that the degenerate case is rather interesting and is treated in well-known papers in Kirchhoff theory, see for example [8]. In the large literature on degenerate Kirchhoff problems, the transverse oscillations of a stretched string, with nonlocal flexural rigidity, depends continuously on the Sobolev deflection norm of u via M( u 2 ). 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.
Throughout the paper we assume that the nonlinear term f : × R → Ris a continuous function, with f (x, t) ≡ 0 for t ≤ 0 and x ∈ . In the following, we also require the following assumptions ( uniformly in ; see [42] for more details; where R 0 is the radius of the largest open ball contained in ; In fact, by a simple calculation, one can verify that , uniformly in x ∈ . Remark 1. 2 We say that f satisfies exponential critical growth at which is reasonable for the nonlinear term f (x, t) behaving like exp(α 0 |t| N /(N −s) ) at infinity. Moreover, by ( f 2 ), for each μ > 0, there exists C μ > 0 such that (1.10)

Remark 1.4
Using ( f 5 ) and the similar discussion as Remark 1.1, one can deduce that for each x ∈ , In For general N ≥ 1 and s ∈ (0, 1), we get the following result. If we consider the special case s = 1/2 and N = 1, then the assumption ( f 6 ) can be removed. Hence we get the second result as follows.
Finally, we consider a special case of f (x, u), that is, we study the following problem: Clearly, g has positive maximum attained at and denote by C r the embedding constant from W s,N /s 0 (1.12) Now we give the third result as follows.
To get the existence of ground state solutions for problem (1.1), we first apply the mountain pass lemma without Palais-Smale condition to get a Palais-Smale sequence {u n } with I (u n ) → c * > 0 and I (u n ) → 0. The main difficulty is how one can get the strong convergence of {u n } and how to prove that the limit of {u n } is the ground state solution of problem (1.1).
To the best of our knowledge, Theorems 1.1-1.3 are the first results for the Kirchhoff-type problems involving critical Trudinger-Moser nonlinearities in the fractional setting.
The rest of the paper is organized as follows. In Sect. 2, we give some necessary properties for the functional setting. In Sect. 3, we verify that the associated functional satisfies the mountain pass geometry and give an estimate for the level value. In Sect. 4, we obtain the existence of ground state solution for problem (1.1). In Sect. 5, a nonnegative and nontrivial solution for problem (1.1) with negative energy is obtained by using Ekeland's variational principle.

Preliminary results
We first provide some basic functional setting that will be used in the next sections. To prove the existence of weak solutions of (1.1), we shall use the following embedding theorem.

Theorem 3.1 Let E be a real Banach space and J
≤ α N ,s and using the fractional Trudinger-Moser inequality, we get Now fix ε > 0 and choose 0 < ρ < ρ 1 < 1 such that ε λ * − C 2 ρ q− θ N s > 0. Thus, Proof It follows from (1.9) that On the other hand, taking μ > θ N /s and using (1.10), we obtain that there exist positive constants C 3 , C 4 > 0 such that for all x ∈ and t ≥ 0.
Hence, I (tu 0 ) → −∞ as t → ∞, thanks to θ N /s < μ. The lemma is proved by taking e = T v 0 , with T > 0 so large that e ≥ ρ and I (e) < 0. Proof Since ψ ≥ 0 in and ψ = 1, as in the proof of Lemma 3.2, we deduce that I (tψ) → −∞ as t → ∞. Consequently, using assumption ( f 6 ), one can deduce that This proves the lemma.
Actually, for the case N = 1 and s = 1/2, assumption ( f 6 ) naturally holds true. To get more precise information about the minimax level c * in this case, let us consider the following Moser functions which have been used in [31]: Let := (a, b), x 0 = a+b 2 and R 0 = b−a 2 . It is standard verify that the functions belongs to W 1 2 ,2 0 ( ). Moreover, lim n→∞ G n = 1 and the support of G n is contained in interval (x 0 − R 0 , x 0 + R 0 ), see [31].
Proof Arguing by contradiction, we assume that Since the functional I possesses the mountain pass geometry, for each n there exists t n > 0 such that In view of the fact that F(x, t) ≥ 0 for all (x, t) ∈ × R, one can deduce that Since M : [0, ∞) → [0, ∞) is a nonnegative function, M is a nondecreasing function. Thus, we get On the other hand, Using change of variable, we have Note that (3.7) implies that It follows from ( f 4 ) that given δ > 0 there exists t δ > 0 such that Thus, there exists n 0 ∈ N such that for all n ≥ n 0 . Hence, which together with (3.7) yields that as n → ∞.
Following some arguments as in [11,13], we are going to estimate (3.8). In view of (3.9), for 0 < δ < β 0 and n ∈ N, we set Splitting the integral (3.8) on U n,δ and V n,δ and using (3.10), we deduce f (x, t n G n )t n G n dx. (3.12) Since G n (x) → 0 a.e. in B R 0 (x 0 ), we deduce that the characteristic functions χ V n,δ satisfies By t n G n < t δ and the Lebesgue dominated convergence theorem, we have as n → ∞ The key point is to estimate the first term on the right hand of (3.12). By (3.7) and the definition of G n , we have for n sufficiently large. Inserting (3.13) and (3.14) in (3.12) and using (3.10), we arrive at Letting δ → 0 + , we obtain which contradicts ( f 4 ). Therefore, the lemma is proved.
By Lemma 3.4, we obtain the desired estimate for the level c * .
Proof Since G n ≥ 0 in and G n → 1, as in the proof of Lemma 3.2, we deduce that Thus, the desired result follows by using Lemma 3.4.
Consider the Nehari manifold associated to the functional I , that is, Proof For any u ∈ N , we define h : [0, +∞) → R by h(t) = I (tu). clearly, h is differentiable and which means that h (1) = 0. Thus, Since c * ≤ c * ≤ I (u 0 ), in order to obtain a ground state u 0 for (1.1) it is enough to show that there is u 0 ∈ A and I (u 0 ) = c * . To this aim, we first give some useful lemmas.
in . Moreover, Thus, the Lebesgue dominated convergence theorem implies that which means that Therefore, (4.4) holds true. By (4.4), ( f 2 ) and the generalized Lebesgue dominated convergence theorem, we have Now, we assert that u = 0. Arguing by contradiction, we assume that u = 0. Then, as n → ∞. Thus, there exists n 0 ∈ N and δ > 0 such that u n N /s < δ < α N ,s α 0 . Choosing q > 1 close to 1 and α > α 0 close to α 0 such that we still have qα u n N /s < δ < α N ,s . Thus, it follows from (2.2) with q = 1 that as n → ∞. Since {u n } is a bounded (P S) c * sequence, we get which implies that From this and assumption (M 1 ), we deduce u n → 0. Furthermore, we obtain I (u n ) → 0, which contradicts the fact that I (u n ) → c * > 0. Therefore, we must have u = 0. So that ξ > 0.
We claim that I (u) ≥ 0. Arguing by contradiction, we assume that I (u) < 0. Set z(t) := I (tu) for all t ≥ 0. Then z(0) = 0 and z(1) < 0. Arguing as in the proof of Lemma 3.1, we can see that z(t) > 0 for t > 0 small enough. Hence there exists t 0 ∈ (0, 1) such that 1] z(t), z (t 0 ) = I (t 0 u), u = 0, which means that t 0 u ∈ N . Therefore, by Remarks 1.1 and 1.4, the semicontinuity of norm and Fatou's lemma, we get By the weak lower semicontinuity of convex functional, we have In view of Remark 1.1 and the continuity of M, we deduce that It follows from above results and (4.5) that which is absurd. Thus the claim holds true. Now we claim that Obviously, by (4.5) and semicontinuity of norm, we have I (u) ≤ c * . Next we are going to show that I (u 0 ) < c * can not occur. Actually, if I (u) < c * , then u < ξ.
Note that (4.5) yields that This gives that On the other hand, by (4.7), we have Thus, it follows from I (u) ≥ 0 that Furthermore, by (M 1 ), we get Note that Hence, it follows from (4.9) that Thus, there exist n 0 ∈ N and α > 0 such that for all n ≥ n 0 . We choose ν > 1 close to 1 and α > α 0 close to α 0 such that In view of (4.8), for some C > 0 and n large enough, we obtain exp(να|u n | N /(N −s) )dx ≤ exp(α |v n | N /(N −s) )dx ≤ C.
Therefore, we deduce from (2. 2) that Define a functional L as follows: In view of the fact that u n → ξ and ξ > 0, by using (M 1 ), we obtain that u n → u in W s,N /s 0 ( ). Furthermore, using (4.5) and the continuity of M , we have I (u) = c * , which is a contradiction. Thus, the assertion (4.6) holds true.
Combining I (u) = c * with I (u n ) → c * and u n → ξ , we conclude that which implies that ξ = u . By the uniform convexity of norm, we obtain that u n → u in W s,N /s 0 ( ). This finishes the proof.
which means that u is a solution of (1.1) satisfying I (u) = c * , that is, I (u) = 0 and I (u) = c * . Therefore, by the definition of c * and c * ≤ c * , we know that u is a ground state solution of problem (1.1). Moreover, Lemma 2.1 shows that u is nonnegative.
where C r > 0 denotes the embedding constant of from W s,N /s 0 ( ) to L r ( ). Since < α N ,s , it follows from the fractional Trudinger-Moser inequality that Let It is easy to check that g has positive maximum attained at Then for λ ≥ * we have t max ≤ ρ 1 < t * . Since
By Lemmas 5.1 and 5.2 and the Ekeland variational principle (see [2]), applied in B ρ λ , there exists a sequence {u n } n such that If ω λ = 0, then by I λ (u n ) → c λ and (4.5) we obtain which is impossible. Thus, we get ω λ > 0. Therefore, from (5.6) and (M 1 ), we conclude that u n − u λ → 0 as n → ∞. In conclusion, the proof is complete.