Multiplicity of nontrivial solutions for a class of fractional Kirchho ff equations

: In this article, we study a class of fractional Kirchho ff with a superlinear nonlinearity:


Introduction and main results
In this article, we investigate the existence of nontrivial solutions for the following fractional Kirchhoff equations with steep potential well: where 0 < α < 1, (−△) α stands for the fractional Laplacian, and f : R N × R → R is continuous.a, λ are positive parameters, b is a nonnegative parameter, 2 * α := 2N N−2α is the critical Sobolev exponent.
Equation (1.1) is related to the stationary analogue of the equation where P 0 , h, E, L are constants, and (1.2) was proposed by Kirchhoff [1] as an extension of D'Alembert's wave equation for free vibrations of elastic strings.Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.In [2], Fiscella and Valdinoci first proposed a stationary fractional Kirchhoff variational model with homogeneous Dirichlet boundary conditions and critical nonlinearity: where M is a continuous Kirchhoff function whose model case is given by M(t) = a + bt.They proved the existence of a solution for the truncated problem.They also obtained the sign of the weak solutions of problem (1.3).There are some interesting results about fractional Kirchhoff equations (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and their references).On the other hand, some studies have focused on the existence and multiplicity of solutions for fractional Kirchhoff equations see [18][19][20][21][22][23][24][25][26].In particular, in [22], they studied the following fractional Kirchhoff equation Under the Berestycki-Lions type assumptions.Applying minimax arguments, they established a multiplicity result for the above equation, provided that q is sufficiently small.In another study [23], the authors studied the existence of multiple solutions for the following fractional p-Kirchhoff equation Applying fibering maps and Nehari manifold, they obtained that the existence of multiple solutions to the above equation for both Hardy-Sobolev subcritical and critical cases.
Peng and Xia (see [24]) considered the existence, multiplicity and concentration of non-trivial solutions of the following concave-convex elliptic equations involving fractional Laplacian: They obtained multiplicity of solutions by applying Nehari manifold decomposition research.
Inspired by some of the previous results, and unlike other literature, we mainly discuss the influence of the number of solutions for functions m and f .We find that the number of differently solutions are obtained when the assumptions about m and f are different.Moreover, we also discuss the existence of ground state solution.
In the next step, we assume the potential function V(x) as follows: (V 1 ) V ∈ C(R N , R) and V ≥ 0 on R N ; (V 2 ) There exists c > 0 such that the set {V < c} := {x ∈ R N |V(x) < c} is nonempty and has finite measure; (V 3 ) Let Ω = intV −1 (0) be a nonempty and smooth boundary with Ω = V −1 (0).(V 1 ) − (V 3 ) are introduced by Bartsch and Wang, please refer to reference [27].
Finally, We can state the following our main results.
The remaining content of this article as follows: Section 2 introduces some preliminary results and some results will be used.Section 3 proving the main results.

Variational setting and preliminaries
We now collect some preliminary results for the fractional Laplacian.A complete introduction of fractional Sobolev space H α (R N ) can be found in [29].
For any α ∈ (0, 1), the fractional Sobolev space H α (R N ) is defined by It is widely known that we endow the space H α (R N ) with the norm and it is also the completion of C ∞ 0 (R N ) with respect to the norm .
The energy functional is defined J µ λ,a on E λ as follows.
Furthermore, it is easy to prove that we can obtain J λ,a ∈ C 1 (H α (R N ), R), and if ϕ is a solution of Eq (1.1), we can get, The following inequalities will be applied to some related theorems.

Proofs of Theorem 1.1-1.4
To complete the proof of Theorem 1.1-1.3,we need the following result.
Lemma 3.1.If assumptions (V 1 ) − (V 3 ) and (F 1 ), (F 2 ) are satisfied.Then for each λ ≥ γ N there eixists Thus, using (2.6) and (3.2), we have, for We can infer that Hence, there exists ∥u∥ H α (R N ),λ = r 0 > 0 small enough, we can get that there exists a constant which is what we wanted to prove.□ By the following Lemma 3.2 and Lemma 3.3, we are able to proof that the functional J µ λ,a satisfies the mountain pass geometry.
Proof.By assumption (F 6 ) and (F 3 ), we have Then, using (1.4) and (3.4), for each u ∈ E and λ ≥ γ N , we obtain So we can infer that Hence, there exists ∥u∥ H α (R N ),λ = r 0 > 0 small enough, we can get that there exists a constant which is what we wanted to prove.
Proof.We set u ∈ E\{0} with u > 0 and define Hence, Using (F 2 ) and Fatou's lemma, we have Therefore, for every a > 0, there exists Applying the above inequality, and by (F 1 ), (F 2 ) − (F 3 ) and Lebesgue's dominated convergence theorem, we obtain Which means that J a,λ (tϕ k ) → −∞ as t → +∞.Hence, there exists v 0 ∈ E with ∥v 0 ∥ H α (R N ),λ > r 0 such that J a,λ (v 0 ) < 0. Thus, we complete the proof.□ Now, we can obtain the following lemma and then finally yield the convergence result.
Then for each Using this, together with conditions (F 1 ), (F 6 ), (F 3 ) and Lebesgue's dominated convergence theorem, yields , there exists v 0 ∈ E with ∥v 0 ∥ H α (R N ),λ > r 0 such that J a,λ (v 0 ) < 0 for all λ > 0. we complete the proof.Lemma 3.6.If assumptions (V 1 ) − (V 3 ), (F 6 ) and (F 3 ) are satisfied.r 0 > 0 is defined as in the proof of Lemma 3.2.Then there exists constants a * > 0 and v 0 ∈ E with ∥v 0 ∥ H α (R N ),λ > r 0 such that J λ,a (v 0 ) < 0 for each 0 < a < a * and λ > 0. Proof.We can get the result as in the proof of Lemma 3.4, so we omit it here.□ Proof of Theorem 1.1 By Lemma 3.1 and 3.3 and the mountain pass theorem [32], we get that for every λ ≥ γ N and a > 0, there exists a sequence {u n } ⊂ E λ such that as n → ∞ where 0 ≤ α a,λ ≤ α a,0 ≤ D a .By the following some lemmas we able to get our main result.
Case II: Using (2.4) and (2.6) leads to In view of (4.3) and (4.6) we have This a contradicts with Case III: for some C * > 0 and for every n.Using (2.4), (2.2) and (F 5 ) it conclude that By (2.4) and the Young inequality we also get that By (4.8) and the fact of In view of (4.7) and (4.9), we can get that By (4.8), we deduce that Using this, together with (4.10), leads to for some C * > 0 and for all n.It follows from (2.2) − (3.1) and condition (F 5 ) that By (4.9) and (4.11) one has We the conclude that the sequence {u n } is bounded in E λ for all constants 0 < a < a * and λ > γ 0 .The proof is complete.
Proof.We may suppose that {u n } is a C α -sequence with α < D. Note that Lemma 4.1, we can get that {u n } is bounded in E λ .Then there exist a subsequence {u n } and u 0 ∈ E λ such that u n ⇀ u 0 weakly in E λ and u n → u 0 strongly in L r loc (R N ) for 2 ≤ r < 2 * α .From this there follows u n → u 0 strongly in Based on the above results and the Hölder inequalities, for each λ > γ N , Let's verify the following conclusion: Case (i)N = 1, 2 : Case (i) N = 1, 2 : Case (ii)N ≥ 3 : Hence, we claim that It is easy see that Υ λ,r → 0 as λ → ∞.Hence we obtain As in the proof of [33], we can get and sup Then by (2.1), (4.14) and Brezis-Lieb Lemma [34], we infer that Furthermore, from the boundedness of the sequence {u n } in E λ we get that there is a number We can infer that there is a number D = D(D > 0) such that By (3.1), (4.13), (4.17) and (4.18) we have which means that there exists γ := γ(a, D) ≥ γ N such that for λ > γ, v n → 0 strongly in E λ .which is what we wanted to prove.□ Lemma 4.4.If assumptions (V 1 ) − (V 3 ), (F 4 ), (F 5 ) with σ ≥ 2α N−2α and (F 2 ) − (F 3 ) are satisfied, N ≥ 3. Then for every D > 0 there exists numbers ), (V 3 ), (F 4 ), (F 5 ) and (F 2 ), (F 5 ) are satisfied, N ≥ 1.Then for any a > 0 and λ > γ, there exists a critical point u λ ∈ E λ of J λ,a (u) such that J λ,a (u λ ) > 0.

Conclusions and discussion of the results
By the condition, f satisfies lim ).We investigated the effects of functions m and Q on the solution.By applying the variational method, we obtain the existence of multiple solutions.Furthermore, it is worth mentioning that the ground state solution has also been obtained.We find that the number of differently solutions are obtained when the assumptions about m and f are different.The main contribution of this paper is to establish a multiplicity theorem in which the main method is based on the variational method.It is worth noting that we have not yet provided multiple solutions for the critical case, and we will continue to study this case.