Nonlocal Schr\"odinger-Kirchhoff equations with external magnetic field

The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.


Introduction and main result
The paper deals with the existence of solutions of the fractional Schrödinger-Kirchhoff problem , M : R + 0 → R + 0 is a Kirchhoff function, V : R N → R + is a scalar potential, A : R N → R N is a magnetic potential, and (−∆) s A is the associated fractional magnetic operator which, up to a normalization constant, is defined as ϕ(x) − e i(x−y)·A( x+y 2 ) ϕ(y) |x − y| N +2s dy, x ∈ R N , along functions ϕ ∈ C ∞ 0 (R N , C). Henceforward B ε (x) denotes the ball of R N centered at x ∈ R N and radius ε > 0. For details on fractional magnetic operators we refer to [12] and to the references [19][20][21][22] for the physical background.
The operator (−∆) s A is consistent with the definition of fractional Laplacian (−∆) s when A ≡ 0. For further details on (−∆) s , we refer the interested reader to [14]. Nonlocal operators can be seen as the infinitesimal generators of Lévy stable diffusion processes [1]. Moreover, they 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 for example [1,5,6] and the references therein). Indeed, the literature on nonlocal fractional operators and on their applications is quite large, see for example the recent monograph [30], the extensive paper [15] and the references cited there.
As stated in [36], up to correcting the operator by the factor (1 − s), it follows that (−∆) s A u converges to −(∇u − iA) 2 u as s ↑ 1. Thus, up to normalization, the nonlocal case can be seen as an approximation of the local case (see Section 2 for further details). As A = 0 and M = 1, equation (1.1) becomes the fractional Schrödinger equation introduced by Laskin [26,27]. Here the nonlinearity f satisfies general conditions. We refer, for instance, to [16,17,34] and the references therein for recent results.
Throughout the paper, without explicit mention, we also assume that A : R N → R N and V : R N → R + are continuous functions, and that V satisfies, The Kirchhoff function M : R + 0 → R + 0 is assumed to be continuous and to verify (M 1 ) for any τ > 0 there exists κ = κ(τ ) > 0 such that M (t) ≥ κ for all t ≥ τ ; and a + b > 0. When M is of this type, problem (1.1) is said to be non-degenerate if a > 0, while it is called degenerate if a = 0. Clearly, assumptions (M 1 ) and (M 2 ) cover the degenerate case. It is worth pointing out that the degenerate case is rather interesting and is treated in well-known papers in Kirchhoff theory, see for example [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, a very realistic model. We refer to [3,7,29,33,38] and the references therein for more details in bounded domains and in the whole space. Recent existence results of solutions for fractional non-degenerate Kirchhoff problems are given, for example, in [18,31,37,39].
Assumptions (M 1 ) and (M 2 ) on the Kirchhoff function M are enough to assure the existence of solutions of (1.1). However, to get the existence of ground states, we assume also the further mild request , where ϑ is the number given in (M 2 ) when (M 2 ) is assumed, otherwise ϑ is any number greater than or equal to 1.
Of course, (M 3 ) is satisfied also in the model case, even when M (0) = 0, that is in the degenerate case. In [33], condition (M 3 ) was also applied to investigate the existence of entire solutions for the stationary Kirchhoff type equations driven by the fractional p-Laplacian operator in R N . Superlinear nonlinearities f satisfy (f 1 ) f ∈ R N × R + → R is a Carathéodory function and there exist C > 0 and p ∈ (2ϑ, 2 * s ) such that ). Now we are in a position to state the first existence result. Sublinear nonlinearities f verify The second result reads as follows. To get infinitely many solutions for equation (1.1) in the local sublinear-superlinear case, we also assume , a nonempty open subset Ω of R N and a 1 > 0 such that An example of f , which satisfies assumptions (f 1 ) and (f 7 ), is  , which is weaker than the coercivity assumption: V (x) → ∞ as |x| → ∞, was first proposed by Bartsch and Wang in [4] to overcome the lack of compactness.
(ii) To our best knowledge, Theorem 1.3 is the first result for the Schrödinger-Kirchhoff equations involving concave-convex nonlinearities in the fractional setting. We also refer to [38] for some related multiplicity results.
The paper is organized as follows. In Section 2 we provide a few remarks about the singular limit as s ↑ 1. In Section 3, we recall some necessary definitions and properties for the functional setting. In Section 4, we obtain some preliminary results. In Section 5, the existence of ground states of (1.1) is obtained by using the mountain pass theorem together with the Nehari method, and by the direct methods respectively. In Section 6, the existence of infinitely many solutions of (1.1) is obtained by using the symmetric mountain pass theorem. 2. Remarks on the singular limit as s ↑ 1 The functional framework investigated in the paper admits a very nice consistency property with more familiar local problems, in the singular limit as the fractional diffusion parameter s approaches 1. Let Ω be a nonempty open subset of R N . We denote by L 2 (Ω, C) the Lebesgue space of complex valued functions with summable square, endowed with the norm u L 2 (Ω,C) . We indicate by H s A (Ω) the space of functions u ∈ L 2 (Ω, C) with finite magnetic Gagliardo semi-norm, given by . Indeed, in the recent paper [36], the following theorem was proved, which is a Bourgain-Brezis-Mironescu type result in the framework of magnetic Sobolev spaces.
and S N −1 is the unit sphere of R N and e any unit vector of R N . Furthermore, Problem (1.1) could be treated in an arbitrary smooth open bounded subset Ω of R N , provided that the solution space is W , which consists of all functions u in H s A (R N ), with u = 0 in R N \ Ω. More precisely, consider the non-degenerate model case where a(s) ≈ 1 − s and b(s) ≈ (1 − s) 2 b 0 as s ↑ 1.
Then the corresponding problem (1.1) in Ω writes as where u belongs to the solution space W and This is natural since the Gagliardo semi-norms are typically multiplied by normalizing constants which vanish at the rate of 1 − s. Since by Proposition 2.1 as s ↑ 1, the above problem converges to the local problem This is the classical model of a Schrödinger-Kirchhoff equation. When b 0 = 0, the last two problems become the classical Schrödinger Dirichlet problems with or without external magnetic potential A.

Functional setup
We first provide some basic functional setting that will be used in the next sections. The critical exponent 2 * The localized norm, on a compact subset K of R N , for the space H s V (K), is denoted by [14,Theorem 6.7], namely there exists a positive constant C such that To prove the existence of radial weak solutions of (1.1), we shall use the following embedding theorem due to P.L. Lions.
Arguing as in [12, Proof. The assertion follows directly from the pointwise diamagnetic inequality Lemma 3.5. Let V satisfy (V 1 ). Let (u n ) n be a bounded sequence in H A,V (R N , C). Then, up to a subsequence, (|u n |) n converges strongly to some function u in L p (R N ) for all p ∈ (2, 2 * s ). Moreover, if V satisfies (V 1 )-(V 2 ), then for all bounded sequence (u n ) n in H s A,V (R N , C) the sequence (|u n |) n admits a subsequence converging strongly to some u in L p (R N ) for all p ∈ [2, 2 * s ).

Preliminary results
The functional I : H A,V (R N , C) → R, associated with equation (1.1), is defined by It is easy to see that I is of class C 1 (H A,V (R N , C), R) and   Proof. Let (u n ) n be a (P S) sequence in H A,V (R N , C). Then there exists C > 0 such that |I(u n )| ≤ C and | I ′ (u n ), u n | ≤ C u n s,A for all n. As in Lemma 4.5 of [7], see also [10], we divide the proof into two parts.
This implies at once that (u n ) n is bounded in H A,V (R N , C), being µ > 2ϑ. Going if necessary to a subsequence, thanks to Lemmas 3.4 and 3.5, we have To prove that (u n ) n converges strongly to u in H A,V (R N , C) as n → ∞, we first introduce a simple notation. Let ϕ ∈ H A,V (R N , C) be fixed and denote by L(ϕ) the linear functional on |f (x, t)t| ≤ ε|t| + C ε |t| p−1 for all x ∈ R N and t ∈ R + .
Using the Hölder inequality, we obtain The Brezis-Lieb lemma and the fact that |u n | → |u| in L p (R N ) give Inserting this in (4.6), we get since ε is arbitrary. Of course, I ′ (u n ) − I ′ (u), u n − u → 0 as n → ∞, since u n ⇀ u in H A,V (R N , C) and I ′ (u n ) → 0 in the dual space of H A,V (R N , C). Thus, this, together with (4.4) and (4.7), implies that If 0 is an isolated point for ([u n ] s,A ) n , then there is a subsequence ([u n k ] s,A ) k such that inf k∈N [u n k ] s,A = d > 0 and one can proceed as before. If, instead, 0 is an accumulation point for ([u n ] s,A ) n , there is a subsequence, still labeled as (u n ) n , such that (4.8) [u n ] s,A → 0, u n → 0 in L 2 * s (R N ) and a.e. in R N .
We claim that (u n ) n converges strongly to 0 in H A,V (R N , C). To this aim, we need only to show that u n 2,V → 0 thanks to (4.8). Now, (4.1) and (4.8) yield that as n → ∞ Hence, (u n ) n is bounded in L 2 (R N , V ) and so in H A,V (R N , C). Thus, by (4.8) and Lemma 3.4 u n ⇀ 0 in H A,V (R N , C) and u n → 0 in L p (R N ), (4.9) being p ∈ (2, 2 * s ). Clearly, by (4.5) and (4.9), for every ε > 0 as n → ∞. Thus, being ε > 0 arbitrary. Obviously, I ′ (u n ), u n → 0 as n → ∞, by (4.9) and the fact that Hence, by the continuity of M and (4.8)-(4.10), we have This shows the claim.
Therefore, I satisfies the (P S) condition in H A,V (R N , C) also in this second case and this completes the proof.

Proof of Theorems 1.1 and 1.2
The following standard Mountain Pass Theorem will be used to get our main result. J(γ(t)).
Then there exists a sequence (u n ) n in E such that J(u n ) → c and J ′ (u n ) → 0 in E ′ , the dual space of E, as n → ∞.
By using (5.1), (M 2 ) and the fact that 2 ≤ 2ϑ < µ, for all u ∈ N , we have Hence, by (M 1 ) and (M 3 ) for u ∈ N where κ = κ(1) > 0 by (M 1 ). Hence in all cases, for all u ∈ N by the elementary inequality t ϑ ≥ t − 1 for all t ∈ R + 0 . In particular, I is coercive and bounded from below on N .
Define c min = inf{I(u) : u ∈ N }. Clearly, 0 ≤ c min ≤ I(u 0 ) = c. Let (u n ) n be a minimizing for c min , namely I(u n ) → c min and I ′ (u n ), u n = 0. Then, since N is a complete metric space, by Ekeland's variational principle we can find a new minimizing sequence, still denoted by (u n ) n , which is a (P S) sequence for I at the level c min . Moreover, Lemma 4.2 implies that (u n ) n has a convergence subsequence, which we still denote by (u n ) n , such that u n → u in H A,V (R N , C). Thus c min = I(u) and I ′ (u), u = 0.
We claim that c min > 0. Otherwise, there is (u n ) n ⊂ H A,V (R N , C) \ {0} with I ′ (u n ) = 0 and I(u n ) → 0. This via (5.2) implies that u n s,A → 0. On the other hand, by (4.11), we have for any ε ∈ (0, V 0 ) s,A . This is a contradiction since 2ϑ < p and proves the claim.
Thus, u is a nontrivial critical point of I, with I(u) = c min > 0. Therefore, u is a ground state solution of (1.1).

Proof of Theorem 1.3
We first recall the following symmetric mountain pass theorem in [23].
Theorem 6.1. Let X be an infinite dimensional real Banach space. Suppose that J is in C 1 (X, R) and satisfies the following condition: (a) J is even, bounded from below, J(0) = 0 and J satisfies the (P S) condition; (b) For each k ∈ N there exists E k ⊂ Γ k such that sup u∈E k J(u) < 0, where Γ k = {E : E is closed symmetric subset of X and 0 / ∈ E, γ(E) ≥ k} and γ(E) is a genus of a closed symmetric set E. Then J admits a sequence of critical points (u k ) k such that J(u k ) ≤ 0, u k = 0 and u k → 0 as k → ∞.
Let h ∈ C 1 (R + 0 , R) be a radial decreasing function such that 0 ≤ h(t) ≤ 1 for all t ∈ R + 0 , h(t) = 1 for 0 ≤ t ≤ 1 and h(t) = 0 for t ≥ 2. Let φ(u) = h( u 2 s,A ). Following the idea of [18], we consider the truncation functional Clearly, I ∈ C 1 (H s A,V (R N , C), R) and . Let (u n ) n be a (P S) sequence, i.e. I(u n ) is bounded and I ′ (u n ) → 0 as n → ∞. Then the coercivity of I implies that (u n ) n is bounded in H s A,V (R N , C). Without loss of generality, we assume that u n ⇀ u in H s A,V (R N , C) and u n → u a.e. in R N . We now claim that Clearly, |f (x, t)t| ≤ C(|t|+|t| p−1 ) for all x ∈ R N and t ∈ R + 0 by (f 1 ). Using the Hölder inequality, we obtain Lemma 3.5 guarantees that |u n | → |u| in L p (R N ) and |u n | → |u| in L 2 (R N ). Hence, u n → u in L p (R N , C) and in L 2 (R N , C) by the Brezis-Lieb lemma. Inserting these facts in (6.2), we get the desired claim (6.1). Now, I ′ (u n ) − I ′ (u), u n − u → 0, since I ′ (u n ) → 0 and u n ⇀ u in H s A,V (R N , C). By (6.1), we have as n → ∞ o(1) = I ′ (u n ) − I ′ (u), u n − u = M ([u n ] 2 s,A ) L(u n ), u n − u − M ([u] 2 s,A ) L(u), u n − u Hence u n → u in H s A,V (R N , C). Case inf n∈N [u n ] s,A = 0. If 0 is an isolated point for ([u n ] s,A ) n , then there is a subsequence ([u n k ] s,A ) k such that inf k∈N [u n k ] s,A = d > 0 and one can proceed as before.
If, instead, 0 is an accumulation point for ([u n ] s,A ) n , there is a subsequence, still labeled as (u n ) n , such that [u n ] s,A → 0 and u n → 0 in L 2 * s (R N ) as n → ∞ and again (6.3) implies at once that u n → 0 in H s A,V (R N , C), since L(u n ) − L(u), u n − u → 0 and M ([u n ] 2 s,A ) → M (0) ≥ 0. In conclusion, I satisfies the (P S) condition in H s A,V (R N , C). For each k ∈ N, we take k disjoint open sets K i such that k i=1 K i ⊂ Ω. For each i = 1, · · · , k let u i ∈ (H s A,V (R N , C) C ∞ 0 (K i , C))\{0}, with u i s,A = 1, and W k = span{u 1 , u 2 , · · · , u k }. Therefore, for any u ∈ W k , with u s,A = ρ ≤ 1 small enough, we obtain by (f 7 ), being q ∈ (1, 2), where C k > 0 is a constant such that u L q (R N ,C) ≤ C k u s,A for all u ∈ W k , since all norms on W k are equivalent. Therefore, we deduce {u ∈ W k : u s,A = ρ} ⊂ {u ∈ W k : I(u) < 0}.
Choosing E k = {u ∈ W k : I(u) < 0}, we have E k ⊂ Γ k and sup u∈Γ k I(u) < 0. Thus, all the assumptions of Theorem 6.1 are satisfied, Hence, there exists a sequence (u k ) k such that I(u k ) ≤ 0, I ′ (u k ) = 0, and u k s,A → 0 as k → ∞.
Therefore, we can take k so large that u k s,A ≤ 1, and so these infinitely many functions u k are solutions of (1.1).