Multiplicity of Small or Large Energy Solutions for Kirchhoff – Schrödinger-Type Equations Involving the Fractional p-Laplacian in R N

We herein discuss the following elliptic equations:M ( ∫ RN ∫ RN |u(x)−u(y)|p |x−y|N+ps dx dy ) (−∆)pu+ V(x)|u|p−2u = λ f (x, u) inRN, where (−∆)p is the fractional p-Laplacian defined by (−∆)pu(x) = 2 limε↘0 ∫ RN\Bε(x) |u(x)−u(y)|p−2(u(x)−u(y)) |x−y|N+ps dy, x ∈ R N . Here, Bε(x) := {y ∈ RN : |x − y| < ε}, V : RN → (0, ∞) is a continuous function and f : RN × R → R is the Carathéodory function. Furthermore,M : R 0 → R+ is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff functionM and the nonlinear term f . The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the L∞-norm.


Introduction
Significant attention has been focused on the study of fractional-type operators in view of the mathematical theory to some phenomena: the social sciences, quantum mechanics, continuum mechanics, phase transition phenomena, game theory, and Levy processes [1][2][3][4][5].
Herein, we discuss the results regarding the existence and multiplicity of nontrivial weak solutions for Kirchhoff-type equations where (−∆) s p is the fractional p-Laplacian operator defined by |x − y| N+ps dy for x ∈ R N , with 0 < s < 1 < p < ∞, ps < N, B ε (x) := {y ∈ R N : |x − y| < ε}, V : R N → (0, ∞) is a continuous function and f : R N × R → R is the Carathéodory function.Furthermore, M ∈ C(R + 0 , R + ) is a Kirchhoff-type function.
Considering the effects of the change in the length of the stings that occurred by transverse vibrations, Kirchhoff in [6] originally proposed the following equation: which is the generalization of the classical D'Alembert's wave equation.Subsequently, most researchers have extensively studied Kirchhoff-type equations associated with the fractional p-Laplacian problems in various ways; see [7][8][9][10][11][12][13][14] and the references therein.The critical point theory, originally introduced in [15] is critical in obtaining the solutions to elliptic equations of the variational type.It is considered that one of the crucial aspects for assuring the boundedness of the Palais-Smale sequence of the Euler-Lagrange functional, which is important to apply the critical point theory, is the Ambrosetti and Rabinowitz condition ((AR)-condition, briefly) in [15].
(AR) There exist positive constants C and ζ such that ζ > p and 0 < ζF (x, t) ≤ f (x, t)t for x ∈ Ω and |t| ≥ C, where F (x, t) = t 0 f (x, s) ds and Ω is a bounded domain in R N .Most results for our problem (1) are to establish the existence of nontrivial solutions under the (AR)-condition; see [7,10,14,16] for bounded domains and [11] for a whole space R N .The (AR)-condition is natural and important to guarantee the boundedness of the Palais-Smale sequence; this condition, however, is too restrictive and gets rid of many nonlinearities.Many authors have attempted to eliminate the (AR)-condition for elliptic equations associated with the p-Laplacian; see [17][18][19][20] and also see [21][22][23][24][25] for the superlinear problems of the fractional Laplacian type.
In this regard, we show that problem (1) permits the existence of multiple solutions under various conditions weaker than the (AR)-condition.In particular, following ([17], Remark 1.8), there exist many examples that do not fulfill the condition of the nonlinear term f in [18,19,21,22,[24][25][26]. Thus, motivated by these examples, the first aim of this paper is to demonstrate the existence of infinitely many large solutions for the problem above using the fountain theorem.One of novelties of this study is to obtain the multiplicity results for problem (1) when f contains mild assumptions different from those of [18,19,21,22,[24][25][26] (see Theorem 1).The other is to demonstrate this result with sufficient conditions for the modified Kirchhoff function M, and the assumption on f similar to that in [18,26] (see Theorem 2).As far as we are aware, none have reported such multiplicity results for our problem under the assumptions given in Theorem 2 of Section 2.
The second aim is to investigate that the existence of small energy solutions for problem (1), whose L ∞ -norms converge to zero, depends only on the local behavior and assumptions on f (x, t), and only sufficiently small t are required.Wang [27] initially investigated that nonlinear boundary value problems admit a sequence of infinitely many small solutions where 0 < q < 1, and the nonlinear term f was considered as a perturbation term.He employed global variational formulations and cut off techniques to obtain this existence result that is a local phenomenon and is forced by the sublinear term.Utilizing the argument in [27], Guo [28] showed that the p-Laplacian equations with indefinite concave nonlinearities have infinitely many solutions.In this regard, lots of authors have considered the results for the elliptic equations with nonlinear terms on a bounded domain in R N ; see [29][30][31].It is well known that the studies in [14,17,19,21,22,26,29,32,33] as well as our first primary result essentially demand some global conditions on f (x, t) for t, such as oddness and behavior at infinity, for applying the fountain theorem to allow an infinite number of solutions.In contrast to these studies that yield large solutions in that they form an unbounded sequence, by modifying and extending the function f (x, t) to a adequate function f (x, t), the authors in [27-29] investigated the existence of small energy solutions to equations of the elliptic type.A natural question is whether the results in [27-31] may be extended to Equation (1).As is known, such a result for Kirchhoff-Schrödinger-type equations involving the non-local fractional p-Laplacian on the whole space R N has not been much studied, although a given domain is bounded.In particular, no results are available even though the fractional p-Laplacian problems without Kirchhoff function M are considered, and we are only aware of paper [34] in this direction.In comparison with the papers [27-29], the main difficulty to obtain our second aim is to show the L ∞ -bound of weak solutions for problem (1).We remark that the strategy for obtaining this multiplicity is to assign a regularity-type result based on the work of Drábek, Kufner, and Nicolosi in [35].Furthermore, it is noteworthy that the conditions on f (x, t) are imposed near zero; in particular, f (x, t) is odd in t for a small t, and no conditions on f (x, t) exist at infinity.This paper is structured as follows.In Section 2, we state the basic results to solve the Kirchhoff-type equation, and review the well-known facts for the fractional Sobolev spaces.Moreover, under certain conditions on f , our problem admits a sequence of infinitely many large energy solutions of our problem (1) via the fountain theorem.Moreover, we assign the existence of nontrivial weak solutions for our problem with new conditions for the modified Kirchhoff function M and the nonlinear term f .In Section 3, we present the existence of small energy solutions for our problem in that the sequence of solutions converges to 0 in the L ∞ -norm.Hence, we employ the regularity result on the L ∞ -bound of a weak solution and the modified functional method.

Existence of Infinitely Many Large Energy Solutions
In this section, we recall some elementary concepts and properties of the fractional Sobolev spaces.We refer the reader to [4,[36][37][38] for the detailed descriptions.Suppose that Let 0 < s < 1 and 1 < p < +∞.We define the fractional Sobolev space W s,p (R N ) by Furthermore, we denote the basic function space W(R N ) by the completion of Following a similar argument in [11,12], we can easily show that the space W(R N ) is a separable and reflexive Banach space.
We recall the continuous or compact embedding theorem in ([11], Lemma 1) and ( [24], Lemma 2.1).Lemma 1.Let 0 < s < 1 < p < +∞ with ps < N.Then, there exists a positive constant C = C(N, p, s) such that, for all u ∈ W s,p (R N ), where p * s = N p N−sp is the fractional critical exponent.Consequently, the space W s,p (R N ) is continuously embedded in L q (R N ) for any q ∈ [p, p * s ].Moreover, the space W s,p (R N ) is compactly embedded in L q loc (R N ) for any q ∈ [p, p * s ).
Lemma 2. Let 0 < s < 1 < p < +∞ with ps < N. Suppose that the assumptions (V1) and (V2) hold.If r ∈ [p, p * s ], then the embeddings We assume that the Kirchhoff function M : R + 0 → R + satisfies the following conditions: where m 0 is a constant.
A typical example for M is given by M(t) = b 0 + b 1 t n with n > 0, b 0 > 0 and b 1 ≥ 0. Next, we consider the appropriate assumptions for the nonlinear term f .Let us denote F (x, t) = t 0 f (x, s) ds and let θ ∈ R be given in (M2).
(F 1) f : R N × R → R satisfies the Carathéodory condition.
Then, it is easily verifiable that A s,p ∈ C 1 (W(R N ), R) and Ψ ∈ C 1 (W(R N ), R).Therefore, the functional E λ is Fréchet differentiable on W(R N ) and its (Fréchet) derivative is as follows: for any u, v ∈ W(R N ).Following Lemmas 2 and 3 in [11], the functional A s,p is weakly lower semi-continuous in W(R N ) and Ψ is weakly continuous in W(R N ).
We now show that the functional E λ satisfies the Cerami condition ((C) c -condition, briefly), i.e., for c ∈ R, any sequence This plays a decisive role in establishing the existence of nontrivial weak solutions.Lemma 3. Let 0 < s < 1 < p < +∞ and ps < N. Assume that (V1), (V2), (M1), (M2), and (F 1)-(F 4) hold.Then, the functional E λ satisfies the (C) c -condition for any λ > 0. where ), it follows from the proceeding as in the proof of Lemma 6 in [11] that {u n } converges strongly to u in W(R N ).Hence, it suffices to verify that the sequence {u n } is bounded in W(R N ).However, we argue by contradiction and suppose that the conclusion is false, i.e., {u n } is a unbounded sequence in W(R N ).Therefore, we may assume that Define a sequence {w n } by Hence, up to a subsequence (still denoted as the sequence {w n }), we obtain w n w in W(R N ) as n → ∞.Furthermore, by Lemma 2, we have for p ≤ r < p * s .Owing to the condition (5), we have The assumption (F 3) implies that there exists t 0 > 1 such that F (x, t) > |t| θ p for all x ∈ R N and |t| > t 0 .From the assumptions (F 1) and (F 2), there is a constant for all x ∈ R N , and for all n ∈ N.
Proof.For c ∈ R, let {u n } be a (C) c -sequence in W(R N ) satisfying (4).Then, relation (5) holds.As in the proof of Lemma 3, we only prove that {u n } is bounded in W(R N ).However, arguing by contradiction, suppose that s by the continuous embedding in Lemma 2. Passing to a subsequence, we may assume that v n v in W(R N ) as n → ∞; then, by compact embedding, v n → v in L r (R N ) for p ≤ r < p * s , and v n (x) → v(x) for almost all x ∈ R N as n → ∞.By the assumption (F 5), one obtains which implies Hence, it follows from the inequality (18) that v = 0. From the same argument as in Lemma 3, we can verify the relations ( 8)- (10), and hence yield the relation (11).Therefore, we arrive at a contradiction.Thus, {u n } is bounded in W(R N ).
Next, based on the fountain theorem in ( [39], Theorem 3.6), we demonstrate the infinitely many weak solutions for problem (1).Hence, we let X be a separable and reflexive Banach space.It is well known that there exists {e n } ⊆ X and { f * n } ⊆ X * such that Let us denote Then, we recall the fountain lemma.
Lemma 5. Let X be a real reflexive Banach space, E ∈ C 1 (X , R) satisfies the (C) c -condition for any c > 0, and E is even.If for each sufficiently large k ∈ N, there exist ρ k > δ k > 0 such that the following conditions hold: Then, the functional E has an unbounded sequence of critical values, i.e., there exists a sequence Using Lemma 5, we demonstrate the existence of infinitely many nontrivial weak solutions for our problem.Theorem 1.Let 0 < s < 1 < p < +∞ and ps < N. Assume that (V1), (V2), (M1), (M2), and (F 1)-(F 4) hold.If f (x, −t) = − f (x, t) satisfies for all (x, t) ∈ R N × R, then the functional E λ has a sequence of nontrivial weak solutions {u n } in W(R N ) such that E λ (u n ) → ∞ as n → ∞ for any λ > 0 .
Proof.Clearly, E λ is an even functional and satisfies the (C) c -condition.Note that W(R N ) is a separable and reflexive Banach space.According to Lemma 5, it suffices to show that there exists for a sufficiently large k.We denote Then, we know α k → 0 as k → ∞.Indeed, suppose to the contrary that there exist ε 0 > 0 and a sequence {u k } in Z k such that Hence, u = 0.However, we obtain which yields a contradiction.For any u ∈ Z k , it follows from assumptions (M2), (F 2), and the Hölder inequality that where C 2 , C 3 and C 4 are positive constants.We choose δ k = (2λC 4 α q k / min{1, m 0 /θ}) 1/(p−q) .Since p < q and α k → 0 as k → ∞, we assert which implies the condition (1).Next, suppose that condition (2) is not satisfied for some k.Then, there exists a sequence {u n } in Y k such that Let As shown in the proof of Lemma 3, we can choose for x ∈ Ω. Considering the inequalities ( 21), ( 22) and Fatou's lemma, we assert by a similar argument to the inequality (10) that Therefore, using the relation ( 23), we have as n → ∞, which yields a contradiction to the relation (20).The proof is complete.
Remark 1.Although we replaced (F 4) with (F 5) in the assumption of Theorem 1, we assert that the problem (1) admits a sequence of nontrivial weak solutions Lastly, we investigate the existence of nontrivial weak solutions for our problem by replacing the assumptions (F 4) and (F 5) with the following condition, which is from the work of L. Jeanjean [40]: When the Kirchhoff function M is constant, and the condition (F 6) with θ = 1 holds, the author in [24] obtained the existence of at least one nontrivial weak solution for the superlinear problems of the fractional p-Laplacian, which is motivated by the works of [18,26].
To the best of our belief, such existence and multiplicity results are not available for the elliptic equation of the Kirchhoff type under the assumption (F 6).Hence, we obtain the following lemma with the sufficient conditions for the modified Kirchhoff function M and the assumption (F 6).Lemma 6.Let 0 < s < 1 < p < +∞ and ps < N. Assume that (V1), (V2), (M1), (M2), (F 1)-(F 3), and (F 6) hold.Furthermore, we assume that , where H(t) = θM(t) − M(t)t for any t ≥ 0 and θ is given in (M2).
Then, the functional E λ satisfies the (C) c -condition for any λ > 0.
Proof.For c ∈ R, let {u n } be a (C) c -sequence in W(R N ) satisfying the convergence (4).Then, the relation ( 5) holds.By Lemma 3, we only prove that {u n } is bounded in W(R N ).Therefore, we argue by contradiction and suppose that the conclusion is false, i.e., In addition, we define a sequence {ω n } by ω n = u n /||u n || W(R N ) .Then, up to a subsequence (still denoted as the sequence {ω n }), we obtain where θ p < q < p * s .We set Ω = x ∈ R N : ω(x) = 0 .From the similar manner as in Lemma 3, we obtain meas(Ω) = 0. Therefore, Let { k } be a positive sequence of real numbers such that lim k→∞ k = ∞ and k > 1 for any k.
> k for a sufficiently large n.Thus, we know by (M2) and the convergence ( 24) that for a large enough n.Then, letting n, k → ∞, we get Since E λ (0) = 0 and E λ (u n ) → c as n → ∞, it is obvious that t n ∈ (0, 1), and E λ (t n u n ), t n u n = 0. Therefore, owing to the assumptions (M3) and (F 6), for all sufficiently large n, we deduce that which contradicts the convergence (25).This completes the proof.
We give an example regarding a function M with the assumptions (M1)-(M3).

Example 1.
Let us see Then, it is easily checked that this function M complies with the assumptions (M1)-(M3).

Existence of Infinitely Many Small Energy Solutions
In this section, we prove the existence of a sequence of small energy solutions for the problem (1) converging to zero in L ∞ -norm based on the Moser bootstrap iteration technique in ( [35], Theorem 4.1) (see also [34]).First, we state the following additional assumptions: (F 7) There exists a constant s 0 > 0 such that pF (x, t) − f (x, t)t > 0 for all x ∈ R N and for 0 < |t| = +∞ uniformly for all x ∈ R N .
Because problem (1) includes the potential term and the nonlinear term f is slightly different from that of [35], a more complicated analysis has to be carefully performed when we apply the bootstrap iteration argument.Proposition 1. Assume that (V1), (M1), and (F 1)-(F 2) hold.If u is a weak solution of the problem (1), Proof.Suppose that u is non-negative.For K > 0, we define , and it follows from the equality (2) that

M(|u|
The left-hand side of the relation ( 26) can be estimated as follows: for some positive constants C 5 and C 6 .Using the assumption (F 2), the Hölder inequality and the relation ( 27), the right-hand side of the relation ( 26) can be estimated: where , and and hence the estimate (28) It follows from relations ( 26), ( 27), (29), and the Sobolev inequality that there exists positive constants C 7 , C 8 and C 9 (independent of K and m > 0) such that To apply the argument that is critical in L ∞ -estimates, we first assume that ||u|| L (m+1)p (R N ) ≥ 1.From the estimate (30), we have which implies for some positive constant C 10 and for any positive constant K, where t is either p or β.The expression in the estimate ( 32) is a starting point for a bootstrap technique.Since u ∈ W(R N ), hence u ∈ L p * s (R N ) and we can choose m := m 1 in the estimate (32) such that (m 1 + 1)t = p * s , i.e., m 1 = p * s t − 1.Then, we have for any positive constant K. Owing to u(x) = lim K→∞ v K (x) for almost every x ∈ R N , Fatou's lemma and the estimate (33) imply Thus, we can choose m = m 2 in the estimate (32) for for some constant C 12 > 0. Meanwhile, we assume that ||u|| L (m+1)p (R N ) < 1.From the relation (30), we have for some positive constant C 13 .Repeating the iterations as in the arguments above, we derive ||u|| L ∞ (R N ) ≤ C 14 for some positive constant C 14 .
If u changes sign, we set positive and negative parts as u + (x) = max{u(x), 0} and u − (x) = min{u(x), 0}.Then, it is obvious that u + ∈ W(R N ) and u − ∈ W(R N ).For each K > 0, we define v K (x) = min{u + (x), K}.Taking again v = v mp+1 K as a test function in W(R N ), we obtain which implies that Proceeding with the similar way as above, we obtain u + ∈ L ∞ (R N ).Similarly, we obtain The following result can be found in [41].Lemma 7. Let E ∈ C 1 (X , R) where X is a Banach space.We assume that E satisfies the (PS)-condition, is even and bounded from below, and E (0) = 0. If, for any n ∈ N, there exist an n-dimensional subspace X n and where S ρ := {u ∈ X : ||u|| X = ρ}, then E possesses a sequence of critical values c n < 0 satisfying c n → 0 as n → ∞.
Based on the work of [27,29], we provide the following two lemmas.
Lemma 8. Assume that (V1), (M1) and (F 1)-(F 2) hold.Furthermore, we assume that M(t) ≤ M(t)t for any t ≥ 0, where M is given in for all x ∈ R N and for t = 0.Then, and It follows from the relations (39) and ( 40) that Consequently, the assumption (38) implies u = 0.
Proof.We can modify and extend the given function f (x, t) to f ∈ C 1 (R N × R, R) satisfying all properties given in Lemma 9. First, we will show that Ẽλ : ) and is even on W(R N ).Moreover, it follows from (F 2) that, for |u(x)| ≤ 2t 0 , there exists a positive constant We set , where t 0 is given in Lemma 9. From the relation (41) and the conditions of κ, we have The above, together with the continuity of the Nemytskij operator with f and acting from L p (B R (0)) into L q (B R (0)), it is clearly shown that the first term on the right side of the inequality (42) tends to 0 as n → ∞.For the second term in the inequality (42), we have From the assumption (F 2), for ε > 0, there exists N(R) ∈ R such that This implies that Ψ is compact in W(R N ), as claimed.Since the derivative of Ψ is compact, it follows from the coercivity of Ẽλ that the functional Ẽλ satisfies the (PS)-condition.The weak lower semicontinuity and the coercivity of Ẽλ ensure that Ẽλ is bounded from below.To utilize Lemma 7, we only need to obtain for any n ∈ N, a subspace X n and ρ n > 0 such that sup X n ∩S ρn Ẽλ < 0. For any n ∈ N, we obtain n independent smooth functions φ i for i = 1, • • • , n, and define X n := span {φ 1 , ..., φ n }.Owing to Lemma 9, when ||u|| W(R N ) <