Nonlocal Kirchhoff Problems with Singular Exponential Nonlinearity

In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular exponential nonlinearity. Under some suitable assumptions, the existence of two nontrivial and nonnegative solutions is obtained by using the mountain pass theorem and Ekeland’s variational principle as the nonlinear term satisfies critical or subcritical exponential growth conditions. Moreover, the existence of ground state solutions for the aforementioned problems without perturbation and without the Ambrosetti–Rabinowitz condition is investigated.


Introduction and Main Results
Let N ≥ 2 and assume that ⊂ R N is a bounded domain with Lipschitz boundary and 0 ∈ . Given s ∈ (0, 1), we study the following fractional Kirchhoff type problem with exponential growth: 2), f : × R → R is a continuous function, and L K is the associated nonlocal integro-differential operator which, up to a normalization constant, is defined as along functions ϕ ∈ C ∞ 0 (R N ). Henceforward B ε (x) denotes the ball of R N centered at x ∈ R N and radius ε > 0. Throughout the paper, we always assume that the singular kernel K : R N \ {0} → R + satisfies the following properties: Obviously, if K(x) = |x| −2N , then L K reduces to the fractional N /s-Laplacian (− ) s N /s . Equations of the type (1.1) are important in many fields of science, notably continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and anomalous diffusion, since they are the typical outcome of stochastically stabilization of Lévy processes, see [2,8,25] 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 [2,8] and the references therein. Indeed, the nonlocal fractional operators have been extensively studied by many authors in many different cases: bounded domains and unbounded domains, different behavior of the nonlinearity, and so on. In particular, many works focus on the subcritical and critical growth of the nonlinearity which allow us to treat the problem variationally using general critical point theory.
Recently, some authors have paid considerable attention in the limiting case of the fractional Sobolev embedding. Let ω N −1 be the N − 1-dimensional measure of the unit sphere in R N and let ⊂ R N be a bounded domain and define W s,N /s 0 ( ) as the completion of C ∞ 0 ( ) with respect to the norm [·] s,N /s which is defined as In [29], Martinazzi  For more details about Trudinger-Moser inequality, we also refer to [23] and [39].
On one hand, in the setting of the fractional Laplacian, Iannizzotto and Squassina in [22] investigated existence of solutions for the following Dirichlet problem (− ) 1 2 u = f (u) in (0, 1), u = 0 i nR \ (0, 1), (1.3) where (− ) 1 2 is the fractional Laplacian and f (u) behaves like exp(α|u| 2 ) as u → ∞. Using the mountain pass theorem, the authors obtained the existence of solutions for problem (1.3). The existence of ground state solutions for (1.3) was discussed in [16]. Subsequently, Giacomoni, Mishra and Sreenadh in [21] studied the multiplicity of solutions for problems like (1.3) by using the Nehari manifold method. For more recent results for problem (1.3) in the higher dimension case, we refer the interested reader to [41] and the references therein. For the general fractional p-Laplacian in unbounded domains, Souza in [13] 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 equation (1.4) by using fixed point theory. Li and Yang [27] studied the following equation where p ≥ 2, 0 < ζ < 1, 1 < q < p, λ > 0 is a real parameter, A is a positive function in L p p−q (R N ), (− ) ζ p is the fractional p-Laplacian and f satisfies exponential growth. On the other hand, Li and Yang in [26] 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 (1.5) as the parameter λ small enough. Mingqi, Rȃdulescu and Zhang studied the following problem where f behaves like exp(α|t| N /(N −s) ) as t → ∞ for some α > 0. Under suitable assumption on M and f , the authors obtained the existence of ground state solutions by using the mountain pass lemma combined with the fractional Trudinger-Moser inequality. 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 established a model governed by the equation 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 ρ 0 is the initial axial tension.  [19] and Miyagaki and Pucci [35]. It is worth mentioning that when s → 1 and M ≡ 1, the equation in problem (1.1) becomes which studied by many authors by using variational methods, see for example, [1,12,15,20,24]. Inspired by the above works, we are devoted to the existence and multiplicity of solutions for problem (1.1) and overcome the lack of compactness due to the presence of critical 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 assumed to be continuous and to verify Recently, the fractional Kirchhoff problems have received more and more attention. Some new existence results of solutions for fractional non-degenerate Kirchhoff problems were given, for example, in [42][43][44]49]. On some recent results concerning the degenerate case of Kirchhoff-type problems, we refer to [3,9,30,45,50] and the references therein. It is worth pointing out that the degenerate case in Kirchhoff theory is rather interesting, for example, it was treated in the seminal paper [11]. 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. Clearly, assumptions (M 1 )-(M 2 ) cover the degenerate case. Define Clearly, by 0 ≤ β < N and the fractional Sobolev embedding, we obtain that λ 1 > 0. First in bounded domain , we assume that the nonlinear term f : × R → R is a continuous function, with f (x, t) ≡ 0 for t ≤ 0 and x ∈ . In the following, we also require the following assumptions ( f 1 )-( f 3 ).
( f 1 ) f satisfies critical growth, i.e., there exists α 0 > 0 such that, and t > 0, and there exists some T > 0 such that and C q 0 > 0 is defined by where C 0 is a positive constant. For the critical case, we have the following result. Let us simply give an sketch of the proofs of Theorems 1.1 and 1.2. Since the problems discussed here satisfies singular exponential growth conditions, the fractional Trudinger-Moser inequality is not available directly. Thus, we first obtain the fractional Trudinger-Moser inequality in singular case. Then, two theorems are proved by using the mountain pass lemma and the Eekand variational principle combined with the singular fractional Trudinger-Moser inequality. To applying the mountain pass theorem and the Ekeland variational principle, we must verify that the associated functional satisfies the Palais-Smale conditions. However, since the nonlinear term satisfies the critical exponential growth, it becomes more difficulty to get the compactness of the energy functional. To overcome the loss of compactness of the energy functional, we have to estimate the range of level value of energy functional. This is the key point to obtain the existence of solutions for the critical case.
Finally, we consider the following problem with critical exponential growth (1.8) To get the existence of ground state solutions for problem (1.8), we also need the following hypotheses: where R 0 is the radius of the largest open ball centered at zero contained in .
In terms of (M 3 ) and Remark 1.1 of [33], we can obtain that In particular, we have Using ( f 6 ) and the same discussion as [33], one can deduce that for each x ∈ , In To get the existence of ground state solutions for problem (1.8), we first show that problem (1.8) has a nonnegative mountain pass solution, and then prove that the mountain pass solution is a ground state solution. The main difficulty is that how we can get the strong convergence of (u n ) n and how to prove that the limit of (u n ) n is the ground state solution of problem (1.8). In the process of proving our main results, some ideas are inspired from papers [17] and [33].
To the best of our knowledge, Theorems 1.1-1.3 are the first results for the Kirchhoff equations involving singular Trudinger-Moser nonlinearities in the fractional setting.
The paper is organized as follows. In Sect. 2, we present the functional setting and show preliminary results. In Sect. 3, by using the mountain pass theorem and Ekeland' variational principle, we obtain the existence of two nontrivial nonnegative solutions for problem (1.1) with subcritical exponential growth conditions as λ small. In Sect. 4, we get the existence of two nonnegative solutions for problem (1.1) with critical exponential nonlinearity. In Sect. 5, we investigate the existence of ground state solutions for problem (1.8) without perturbation term and the Ambrosetti-Rabinowitz condtion.

Preliminary Results
In this section, we give the variational framework of problem (1.1) and prove several necessary results which will be used later.
here we apply (k 1 ). By a similar discussion as in [44], we know that (W For ν ≥ 1 and β < N , we define To prove the existence of weak solutions for problem (1.1), we shall use the following embedding theorem.
Proof By [33], we know that the embedding W s,N /s 0 Next we show that W s,N /s 0 To this aim, we choose t > 1 close to 1 such that βt < N . Then for any bounded sequence (u n ) n in W s,N /s 0 This proves the theorem.

Theorem 2.3 Let N ≥ 2 and let be a bounded domain in
Proof Choose σ > 1 such that σβ < N . Then by the Hölder inequality and the fractional Trudinger-Moser inequality, we have Then by the Hölder inequality and the fractional Trudinger-Moser inequality, we deduce that Now we define the Moser functions which have been used in [40]: By the result in [40], we get Moreover, we have In conclusion, the proof is complete.
We give a singular fractional version of theorem of P.L. Lions ( [28]).

and converging weakly to a nonzero function u. Then for any
Proof By the Hölder inequality, we obtain Therefore, the desired result holds true.
To study the nonnegative solutions of problems (1.1) and (1.8), we define the associated functionals I λ , I : W s,N /s Since f is continuous and satisfies 3) is arbitrary. Using (2.3), Theorem 2.3 and the assumption on K, one can verify that the functionals I λ and I are well defined, of class Observe that for a.e. x, y ∈ , x ∈ by assumption and h(−u − λ ) ≤ 0 a.e. in . Hence, This, together with u λ > 0 and (M 1 ), implies that u − λ ≡ 0, that is u λ ≥ 0 a.e. in . Similarly, one can verify that any nontrivial critical point of functional I is nonnegative.

The Subcritical Case
Let us recall that I λ satisfies the (P S) c condition in W s,N /s namely a sequence such that I λ (u n ) → c and I λ (u n ) → 0 as n → ∞, admits a strongly convergent subsequence in W s,N /s 0,K ( ).
In the sequel, we shall make use of the well-known mountain pass theorem. For the reader's convenience, we state it as follows (see for example [46]).

Theorem 3.1 Let X be a real Banach space and J
(ii) there exists e ∈ X satisfying e X > ρ such that J (e) < 0.
J (γ (t)) ≥ α and there exists a (P S) c sequence (u n ) n ⊂ X.
To find a mountain pass solution of problem (1.1), let us first verify the validity of the conditions of Theorem 3.1. Proof Since f satisfies subcritical growth condition ( f 1 ), for q > θ N /s and any α > 0, we have for all u ∈ W s,N /s 0,K ( ) and any ε ∈ (0, λ 1 ). Since 0 ≤ β < N , we can choose ν > 1 close to 1 such that βν < N . It follows from Theorem 2.1 and (K 2 ) that there exists C > 0 such that Thus, we deduce from (3.1) that Fix ε ∈ (0, λ 1 ) and define Due to θ N /s < q, we can choose 0 < ρ ≤ ρ 1 < 1 such that g(ρ) > 0. Thus, On the other hand, using ( f 2 ) and the continuity of f , there exist positive constants thanks to θ N /s < μ. The lemma is proved by taking e = T 0 v 0 , with T 0 > 0 so large that e ≥ ρ and I λ (e) < 0.  Otherwise, 0 is an accumulation point of the sequence ( u n ) n and so there exists a subsequence (u n k ) k of (u n ) n such that u n k → 0 strongly in W s,N /s 0,K ( ) as k → ∞. Thus, we need only consider the case d := inf n≥1 u n > 0.
In the following, we assume that d := inf n≥1 u n > 0. We first show that (u n ) n is bounded in W s,N /s 0,K ( ). Using (M 1 ), (M 2 ) and ( f 2 ) with μ > θ N s , we get as n → ∞, thanks to Theorem 2.2. Thus, (3.8) holds true.
Since (u n ) n is a bounded (P S) c sequence, we get as n → ∞ which implies that Moreover, one can prove that u, u n − u s,N /s → 0. Hence we obtain that By using a similar discussion as [33], we have u n → u in W s,N /s 0,K ( ). This ends the proof.
Proof of Theorem 1.1 By Lemmas 3.1 and 3.2, we know that there exists a threshold λ * > 0 such that for all 0 < λ < λ * , I λ satisfies all the assumptions of Theorem 3.1. Hence there exists a (P S) c sequence. Moreover, by Lemma 3.3, for all λ < λ * the functional I λ admits a nontrivial critical point u 1 ∈ W s,N /s 0,K ( ). The critical point u 1 is a nontrivial mountain pass solution of problem (1.1). Furthermore, Lemma 2.1 shows that u 1 is nonnegative.
Next we show that problem has another nontrivial and nonnegative solution. Define where ρ > 0 is given by Lemma 3.1 and B ρ = {u ∈ W s,N /s 0,K ( ) : u < ρ}. Now we claim that c λ < 0. Consider the following problem By the direct method and 0 ≤ h ∈ (W s,N /s 0,K ( )) * , one can verify that the above problem has a unique nonnegative solution v ∈ W s,N /s for all 0 ≤ t ≤ 1 small enough. Since N /s > 1, it follows that I λ (tv) < 0 for t ∈ (0, 1) small enough. Thus, the claim is true. By Ekeland's principle and a standard argument, there exists a sequence (u n ) n ⊂ B ρ such that I λ (u n ) → c λ < 0 and I λ (u n ) → 0 as n → ∞. Furthermore, Lemma 3.3 yields that (u n ) n converges to some u 2 strongly in W s,N /s 0,K ( ), and so u 2 is a nontrivial and nonnegative solution of problem (1.1). Clearly, u 1 and u 2 are two distinct solutions.

The Critical Case
In this section, we consider the critical case of problem (1.1). Without further mentioning, we always assume that f satisfies ( f 1 ), ( f 2 ) − ( f 4 ), and M(t) = t θ−1 with θ > 1. To prove Theorem 1.2, we first give several necessary results.  We first consider c > 0. By ( f 2 ) and the assumption on M, we have which means that (u n ) n is bounded in W s,N /s 0,K ( ). Thus, we get For any ε > 0, by the Young inequality we have Taking ε = 1 2 s N θ − 1 μ in above inequality and putting it into (4.1), we obtain 1 2 Then for all 0 < λ < 3 , we get lim sup . If c < 0, then with a similar discussion as above, one can easily get that there exists 3 > 0 such that the (P S) sequence satisfies (4.2). Therefore, there exists 3 = min{ 3 , 3 } such that (4.2) holds true. It follows from (4.2) that there exist n 0 ∈ N and δ > 0 such that u n N /(N −s) < δ < N −β N α N ,s α 0 . Choosing ν > 1 close to 1 and α > α 0 close to α 0 such that we still have να u n N /(N −s) < δ < N −β N α N ,s . Thus, it follows from (2.2) with q = 1 that Clearly, C q 0 > 0. By Theorem 2.2, one can easily verify that there exists a nonnegative function ϕ 0 ∈ W s,N /s In view of the proof of Lemma 4.1, we take γ (t) = t T ϕ 0 , where T > 0 is sufficiently large such that e = T ϕ 0 . Hence, it follows from the definition of c 1 that which implies that Furthermore, from ( f 4 ), we obtain By the assumption on C 0 , (4.3) holds. Thus, it follows from Lemma 4.2 that there exists 4 = min{ 2 , 3 } such that problem (1.1) has a nontrivial nonnegative solution.
To show that problem has another solution, we set where ρ 2 > 0 is given by Lemma 4.1 and With a similar discussion as the proof of Theorem 1.1, we can prove that c 2 < 0. By Lemma 4.1, we obtain By Ekeland's variational principle, there exists a sequence (v n ) n ⊂ B ρ 2 such that Thus, u λ is a nontrivial nonnegative solution with I λ (u λ ) < 0. Thus, the proof is complete.
The following version of the mountain pass theorem, which will be used later, shows us the existence of a Cerami sequence at the mountain pass level. Theorem 5.1 (See [10]) Let X be a real Banach space with its dual space E * and assume that J ∈ C 1 (X , R) satisfies for some , σ, ρ > 0 and e ∈ X with e X > ρ. Let c be characterized by Then there exists a Cerami sequence (u n ) n in X , that is, To this aim, let us first verify the validity of the conditions of Theorem 5.1.

where G n is given by Theorem 2.3.
Proof Arguing by contradiction, we assume that Since I possesses the mountain pass geometry, for each n, max t≥0 I (tG n ) is attained at some t n > 0, that is, we deduce Next we show that (t n ) n is bounded. Using change of variable, we deduce from (5.4) that It follows from ( f 5 ) that given δ > 0 there exists t δ > 0 such that  as n → ∞. Inspired by [12,17,33], we are going to estimate (5.4). In view of (5.5), for 0 < δ < β 0 and n ∈ N, we set Splitting the integral (5.4) on U n,δ and V n,δ and using (5.5), we deduce 1 |x| β f (x, t n G n )t n G n dx. (5.8) Since G n (x) → 0 a.e. in B R 0 (0), we deduce that the characteristic functions χ V n,δ satisfies χ V n,δ → 1 a.e. in B R 0 (0) as n → ∞.
By t n G n < t δ and the Lebesgue dominated convergence theorem, we have The key point is to estimate the first term on the right hand of (5.8). By (5.3) and the definition of G n , we have (5.10) Inserting (5.9) and (5.10) in (5.8) and using (5.7), we arrive at Letting δ → 0 + , we obtain which contradicts ( f 5 ). Therefore, the lemma is proved.
By Lemma 5.3, 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 5.2, we deduce that I (tG n ) → −∞ as t → ∞. Consequently, Thus, the desired result follows by using Lemma 5.3.
Consider the Nehari manifold associated to the functional I , that is, Proof The proof is similar to [17] and [33], so we omit the proof. If d := inf n≥1 u n = 0, we can discuss as Lemma 3.3. Thus, we need only consider the case d := inf n≥1 u n > 0.
In the following, we assume that d := inf n≥1 u n > 0. We first show that (u n ) n is bounded in W s,N /s 0,K ( ). Arguing by contradiction, we assume that u n ≥ 1 and lim n→∞ u n = ∞.
Set v n = u n u n .
Then v n = 1. Going if necessary to a subsequence, we can assume that v n v in W s,N /s 0,K ( ). Further, one can show that Now we prove that v + = 0 a.e. in . If the Lebesgue measure of set U + := {x ∈ : v + (x) > 0} is positive, then we have Thus, by (1.10), we deduce which implies that lim n→∞ F(x, u + n (x)) |x| β |u + n | N θ/s = ∞ a.e. in U + .
It follows that lim inf n→∞ F(x, u + n (x)) |x| β |u + n | N θ/s dx = ∞. (5.11) Note that (u n ) n is a Cerami sequence at level c * . Then which together with lim t→∞ M (t) = ∞ yields that Here we have used the fact that It follows that Since v + On the other hand, by u n → ∞, we deduce Thus, letting n → ∞ and then letting R Since I (0) = 0 and I (u n ) → c * , we can assume that t n ∈ (0, 1). Then d dt I (tu n )| t n = 0. Thus, we get I (t n u n ), t n u n = 0, that is, From (1.11), it yields that Moreover, it follows from (u n ) n is a Cerami sequence that (5.14) Now, we assert that u = 0. Arguing by contradiction, we assume that u = 0. Then, 1 |x| β F(x, u n )dx → 0 and I (u n ) → c gives that as n → ∞. Thus, there exist n 0 ∈ N and δ > 0 such that Choosing ν > 1 close to 1 and α > α 0 close to α 0 such that we still have να u n N /(N −s) < δ < N −β N α N ,s . Thus, it follows from (2.2) with q = 1 that as n → ∞. Since (u n ) n is a bounded Cerami sequence, we get which implies that From this and assumption (M 1 ), we deduce u n → 0, which contradicts the assumption that inf n≥1 u n > 0. Therefore, we must have u = 0. We claim that I (u) ≥ 0. Arguing by contradiction, we assume that I (u) < 0. Set ζ(t) := I (tu) for all t ≥ 0. Then ζ(0) = 0 and ζ(1) < 0. Arguing as in the proof of Lemma 3.1, we can see that ζ(t) > 0 for t > 0 small enough. Hence there exists t 0 ∈ (0, 1) such that 1] ζ(t), ζ (t 0 ) = I (t 0 u), u = 0, which means that t 0 u ∈ N . Therefore, by Remarks 1.1 and 1.2, the semicontinuity of norm and Fatou's lemma, we get By the weak lower semicontinuity of convex functional, we have Obviously, by (5.14) and semicontinuity of norm, we have I (u) ≤ c * . Next we prove that I (u 0 ) < c * can not occur. Actually, if I (u) < c * , then u < ξ.
Note that (5.14) yields that In view of (5.17), for some C > 0 and n large enough, we obtain 1 |x| β exp(να|u n | N /(N −s) )dx ≤ 1 |x| β exp(α |w n | N /(N −s) )dx ≤ C. Therefore, we deduce from (2.2) that 1 |x| β f (x, u n )(u n − u)dx ≤ C 1 |x| β |u n − u| N θ/s dx + In view of the fact that u n → ξ and ξ > 0, by using (M 1 ) and a similar discussion as in [33], we obtain that u n → u in W s,N /s 0,K ( ). Furthermore, using (5.14) and the continuity of M , we have I (u) = c * , which is a contradiction. Thus, the assertion (5.15) holds true.
Combining I (u) = c * with I (u n ) → c * and u n → ξ , we conclude that which means that u is a nontrivial solution of (1.8) 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 (1.8). Moreover, Lemma 2.1 shows that u is nonnegative.