On a class of Kirchhoff equations involving an anisotropic operator and potential

In this work, we are concerned with a class of fractional equations of Kirchhoff type with potential. Using variational methods and a variant of quantitative deformation lemma, we prove the existence of a least energy sign-changing solution. Moreover, the existence of infinitely many solution is established.


Introduction and main result
Consider the following fractional Kirchhoff equation with C N ,α is a normalization constant and P.V. stands for the Cauchy principal value; and V, f are a functions satisfying some conditions which will be specified later. The presence of the nonlocal term [u] 2 α in (1.1) causes some mathematical difficulties and so the study of such a class of equations is of much interest. Moreover, Eq. (1.1) is a fractional version related to the following hyperbolic equation which was proposed by Kirchhoff [22] as an extension of the classical D'Alembert's wave equation by considering the changes in the length of the strings produced by transverse vibrations. On the other hand, when b = 0, V ≡ 1 and f (u) = u p , equation (1.1) appears in the study of solitary waves of the generalized Benjamin-Ono-Zakharov-Kuznetsov equation see [28] for some local and global well-posedness results on this equation with m = n = 1. The anisotropic operator (− x ) α u − y u is observed in the study of toy models parabolic equations for which local diffusions occur only in certain directions and nonlocal diffusions. It models diffusion sensible to the direction in the Brownian and Lévy-Itô processes. For some regularity and rigidity properties of this operator, the readers can refer to [6,17]. Recently, Esfahani [15] considered Eq. (1.1) with b = 0. Under suitable assumptions on f ∈ C 1 (R, R), by adapting some arguments developed in [7][8][9] and using a variant of deformation lemma, the author shows that the equation admits a least energy sign-changing solution.
It is interesting to note that the fractional Laplacian (isotropic operator) problems have attracted a great attention by various study on the existence and multiplicity of solutions; see, e.g., [19,20,25,33]. This interest comes from its significant applications in several areas such as physics, biology, chemistry and finance; see, e.g., [5,11,12,18,23]. We also refer to [10] for more applications specifically to an audience of mathematicians. We point out that the variational methods leading to existence theories in the fractional setting were introduced in [30].
In [18], Fiscella and Valdinoci first introduced a stationary Kirchhoff variational equation, which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. After that, the fractional Kirchhoff problems have been investigated by many researchers, we cite here [2,3,13,14,21,26,27,31]. However, to the best of our knowledge, there are no result about existence of solutions for a Kirchhoff-type equation with an operator containing local and nonlocal diffusions except [24]. Inspired by the results mentioned above, especially [1,4,15,21], in the present paper, we will prove the existence of sign-changing solutions and infinitely many solutions for Eq. (1.1). To this aim, we need a compact embedding which is appropriate to our equation and some technical lemmas related to the minimization method on the nodal Nehari manifold.
We assume that V : R N → R is continuous and satisfies: For the nonlinearity f, we assume that f ∈ C 1 (R, R) and Before stating what we think our main result, we recall our working Sobolev space. Let H α (R N ) be the fractional Sobolev-Liouville space We search the solutions in the following subspace and its related norm A sign-changing solution of (1.1) is a weak solution u ∈ E α satisfying u ± = 0, where u + = max(u, 0) and u − = min(u, 0).
The main result can be described by the following theorem.

Auxiliary results and proof of main theorem
In this section, we give some preliminary results for the proof of Theorem 1.1.  [15]) Assume that (V 0 ) holds. Then, E α is continuously embedded into L q (R N ) for q ∈ [2, p * ]; and compactly embedded into L q (R N ) for q ∈ [2, p * ); Obviously, the energy functional associated to (1.1) defined on E α by is of class C 1 and its critical points are solutions of (1.1).
To discus the existence of sign-changing solutions, we follow some arguments developed in [7][8][9]32]. More precisely, we will minimize the functional I on the nodal set  i) there exists an unique t u > 0 such that g u (t u ) = 0, g u > 0 on (0, t u ) and g u < 0 on (t u , ∞); (ii) ρ := inf u∈S 1 g u (t u ) > 0. Furthermore, for each compact ⊂ S 1 , there exists C > 0 such that t u ≤ C for all u ∈ ; (iii) let : E α \{0} → N and : S 1 → N be the mappings defined by Then, is continuous and is a homeomorphism. Moreover, the inverse of is given by −1 (u) = u ||u|| Eα .
Therefore, by Lemma 2.2, for all t > 0, (2.2) It follows from Fatou's lemma and (f 3 ), Hence, for some T > 0 large enough, Moreover, using (f 4 ), it easy to see that t u is the unique number satisfying g u (t u ) = 0, and hence (i) follows.
. Then ρ > 0. Indeed, by (2.1), we can find t > 0 such that Thus ρ > 0. Now, let ⊂ S 1 be a compact set. Suppose by contradiction that there exists {u n } ⊂ such that t u n → +∞. Then, up to subsequence, u n → u for some u ∈ . By (2.3), we have lim n→∞ I (t u n u n ) = −∞. (2.5) is increasing on (0, ∞) and decreasing on (−∞, 0). Therefore Since I (t u n u n ), t u n u n = 0 for all n ∈ N, This contradicts with (2.5).
Letting n goes to infinity, we obtain which yields that l 0 u ||u|| Eα ∈ N and t u = l 0 ||u|| Eα . Consequently (u n ) → (u), and so is continuous. The proof of Lemma 2.3 is completed.
Consider the functionals : E α \{0} → R and : S 1 → R given by In view of Lemma 2.3 and with the help of [32], we have the following result.
Proof The proof is similar to that of [32, Proposition 9, Corollary 10].
(a 2 ) For s large enough, we have ξ 1 (s) < s. If not, there is s n → ∞ such that ξ 1 (s n ) ≥ s n for all n. Then ξ 1 (s n ) → ∞. Therefore, using (2.16), we arrive at a contradiction and consequently (a 2 ) follows. The claim is proved.

Proof of the Claim 2.8
Since Without loss of generality, we may assume that 0 < t 0 ≤ s 0 . Then The first part of (2.18) can be written Dividing by t 4 0 , we obtain On the other hand, v If t 0 < 1, by (f 4 ) and (2.21) we derive a contradiction. Then s 0 ≥ t 0 ≥ 1. Similarly, using the second part of (2.18), we get 4 dxdy. (2.22) This and (f 4 ) imply t 0 ≤ s 0 ≤ 1. So, we conclude that t 0 = s 0 = 1. The claim is proved.
Then v ± 1 = 0 and Therefore, as a consequence of (i), t 2 t 1 , s 2 s 1 is a critical point of ϕ v 1 . From Claim 2.8, we obtain t 2 t 1 = s 2 s 1 = 1, that is t 1 = t 2 and s 1 = s 2 , and hence the uniqueness is achieved.
Next, we will show that ϕ u admits a global maximum. (2.23) Observing that Hence, ϕ u (t, s) → −∞ as |(t, s)| → ∞. So, by the continuity, we conclude that ϕ u admits a global maximum This together with the fact that max t≥0 ϕ u (t, 0), max s≥0 ϕ u (0, s) > 0 leads to which shows thatt,ŝ > 0. Consequently (t,ŝ) is a critical point with positive coordinates. By the uniqueness of (t + , s − ), we deduce that (t,ŝ) = (t + , s − ), and hence (t + , s − ) is an unique global maximum. This finishes the proof of Lemma 2.6.

Lemma 2.9
Let {u n } ⊂ M. If u n u weakly in E α , then u ± = 0.
Proof For all v ∈ M, we have Therefore, using (f 1 )-(f 2 ) and Lemma 2.2, for ε > 0, there exists C ε , such that Let u n u weakly in E α . It follows from (2.24) that which implies that u ± = 0. Let us first prove that {u n } is bounded in E α . Suppose that up to a subsequence ||u n || E α → ∞ and set v n = u n ||u n || Eα . Then {v n } is bounded in E α . So, in view of Lemma 2.2, we may assume that Therefore, for all t > 0 and n ∈ N, This is impossible and hence {u n } is bounded in E α . Then u n u in E α . From Lemma 2.9, we infer that u ± = 0. Applying Lemma 2.6, there exist t + , s − > 0 such that t + u + + s − u − ∈ M and Since u n u and u n ∈ M, it follows from (2.25) and Fatou's lemma that Similar to the proof of Claim 2.8, we see that t + , s − ≤ 1. It follows from (f 4 ) and (2.25)-(2.26) that This yields Arguing as in the proof of Claim 2.8, we derive that t + = s − = 1, and so Now, we show that I (u) = 0. Suppose by contradiction that I (u) = 0. Then, by the continuity of I , there exist ν, > 0 such that In view of (2.31) and [34,Lemma 2.3], for ε < min ν 8 , We claim that max (t,s)∈D Indeed, by Lemma 2.6, On the other hand, we have φ 0 (t, s) = I (γ u (t, s)), u + , I (γ u (t, s)), u − =: φ 1 0 (t, s), φ 2 0 (t, s) and By direct computation and Similarly, we have Therefore, Jac(φ 0 )(1, 1) > 0.
Note that (1, 1) is the unique pre-image of (0, 0) by φ 0 , thus (1, 1) is the unique regular point of φ 0 . By the Brouwer degree theory, deg(φ 0 , D, (0, 0)) = sign(Jac(φ 0 )(1, 1)) = 1. but this contradicts (2.33). We conclude that u is a critical point of I and so it is a sign-changing solution of (1.1). If f is odd, then I is even. In view of Lemma 2.4 and Remark 2.5, is bounded below in S 1 . By standard arguments and Lemma 2.3, we see that satisfies (PS) condition on S 1 . According to [29,Theorem 8.10] and Lemma 2.4, we deduce that I has infinitely many critical points, which completes the proof of Theorem 1.1.
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