Concentration results for a magnetic Schrödinger-Poisson system with critical growth

Abstract: This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation  ( ε i ∇ − A(x) )2 u + V(x)u + ε−2(|x|−1 * |u|2)u = f (|u|2)u + |u|4u in R3, u ∈ H1(R3,C), where ε > 0, V : R3 → R and A : R3 → R3 are continuous potentials, f : R → R is a subcritical nonlinear term and is only continuous. Under a local assumption on the potential V, we use variationalmethods, penalization technique and Ljusternick-Schnirelmann theory to prove multiplicity and concentration of nontrivial solutions for ε > 0 small.


Introduction and main results
In this paper, we study multiplicity and concentration of the nontrivial solutions of the following Schrödinger-Poisson type equations with critical growth ϵ i ∇ − A(x) u + V(x)u + ϵ − (|x| − * |u| )u = f (|u| )u + |u| u in R , (1.1) where u ∈ H (R , C), ϵ > is a parameter, V : R → R is a continuous function, f ∈ C(R, R) has a subcritical growth, the magnetic potential A : R → R is Hölder continuous with exponent α ∈ ( , ], and the convolution potential is de ned by |x| − * |u| = R |x − y| − |u(y)| dy. In recent years a considerable amount of work has been devoted to investigating the existence and multiplicity of solutions for nonlinear Schrödinger-Poisson system without magnetic eld. We notice that, by using minimax theorema and the Ljusternik-Schnirelmann theory, He [25] gave multiplicity and concentration of positive solutions of the following problem where f ∈ C (R) has the subcritical growth and the potential V satis es a global condition introduced by Rabinowitz [31]. In [26], He and Zou studied the existence and concentration of ground state solutions for the following Schrödinger-Poisson system with the critical growth − ϵ ∆u + V(x)u + ϕ(x)u = f (u) + |u| u, in R , − ϵ ∆ϕ = u , u(x) > , in R , (1.2) where f ∈ C (R) and the potential V satis es a global condition. Then, He [27] studied the multiplicity of concentrating positive solutions for Schrödinger-Poisson system (1.2) with nonlinear term f ∈ C(R) under a local assumption introduced by del Pino and Felmer [17]. For further results on Schrödinger-Poisson system without magnetic eld, we refer to [1,4,5,14,15,22,32,33,36,40] and the references therein(see also [21] for the fractional case).
Concerning the magnetic nonlinear Schrödinger equation (1.1), we refer to [6-8, 10-13, 16, 19, 23, 24, 29, 38, 39] and references therein. It is well known that the rst result involving the magnetic eld was obtained by Esteban and Lions [19]. They used the concentration-compactness principle and minimization arguments to obtain solutions for ε > xed. In [39], the authors studied multiplicity and concentration of solutions for magnetic relativistic Schrödinger equations, Xia [38] studied a critical fractional Choquard-Kirchho problem with magnetic eld. In particular, due to our scope, we want to mention [41] where the authors studied a Schrödinger-Poisson type equation with magnetic eld by using the method of the Nehari manifold, the penalization method and Ljusternik-Schnirelmann category theory for subcritical nonlinearity f ∈ C . If f is only continuous, then the arguments in [41] failed. Recently, by variational methods, penalization technique, and Ljusternick-Schniremann theory, for the magnetic Schrödinger-Poisson system with subcritical growth nonlinearity f which is only continuous, in [29] we proved multiplicity and concentration properties of nontrivial solutions for ϵ > small. For the fractional Schrödinger-Poisson type equations with magnetic eld, we refer to [2,3].
Inspried by [27,29], we intend to prove multiplicity and concentration of nontrivial solutions for problem (1.1) with critical growth. Since the probem we deal with has the critical growth, we need more re ned estimates to overcome the lack of compactness. On the other hand, due to the appearance of magnetic eld A(x) and the nonlocal term |x| − * |u| , problem (1.1) will be more di cult, and some estimates are also more complicated.
In this paper, we make the following assumptions on the potential V: (V )There exists V > such that V(x) ≥ V for all x ∈ R ; (V )There exists a bounded open set Λ ⊂ R such that On the nonlinearity f ∈ C(R, R), we require that: (f ) f (t) t is strictly increasing in ( , ∞). The main result of this paper is listed as follows: Theorem 1.1. Assume that V satis es (V ), (V ) and f satis es (f )-(f ). Then, for any δ > such that there exists ϵ δ > such that, for any < ϵ < ϵ δ , problem (1.1) has at least cat M δ (M) nontrivial solutions. Moreover, for every sequence {ϵn} such that ϵn → + as n → +∞, if we denote by uϵ n one of these solutions of (1.1) for ϵ = ϵn and ηϵ n ∈ R the global maximum point of |uϵ n |, then The paper is organized as follows. In Section 2 we indicate the functional setting and give some preliminary results. In Section 3, we study the modi ed problem, and prove the Palais-Smale condition for the modi ed functional and provide some tools which are useful to establish a multiplicity result. In Section 4, we study the autonomous limit problem associated. It allows us to show the modi ed problem has the multiple solutions. Finally, the proof of Thereom 1.1 is derived in Section 5.

Notation
• C, C , C , . . . denote any positive constants, whose exact values are not relevant; • B R (y) denotes the open disk centered at y ∈ R with radius R > and B c R (y) denotes the complement of B R (y) in R ; • · , · q, and · L ∞ (Ω) denote the usual norms of the spaces H (R , R), L q (R , R), and L ∞ (Ω, R), respectively, where Ω ⊂ R . ·, · denotes the inner product of the space H (R , R).

Abstract setting and preliminary results
For u : R → C, let us denote by The space H A (R , C) is an Hilbert space endowed with the scalar product where Re and the bar denote the real part of a complex number and the complex conjugation, respectively. Moreover we denote by u A the norm induced by this inner product.
On H A (R , C), an important tool is the following diamagnetic inequality (see e.g. [28,Theorem 7.21]) Now, by a simple change of variables, we can see that (1.1) is equivalent to where Aϵ(x) = A(ϵx) and Vϵ(x) = V(εx). Let Hϵ be the Hilbert space obtained as the closure of C ∞ c (R , C) with respect to the scalar product u, v ϵ := Re R ∇ Aϵ u∇ Aϵ v + Vϵ(x)uv dx and let us denote by · ϵ the norm induced by this inner product. By the diamagnetic inequality (2.1), we have, if u ∈ H Aϵ (R , C), then |u| ∈ H (R , R) and u ≤ C u ϵ. Therefore, the embedding Hϵ → L r (R , C) is continuous for ≤ r ≤ and the embedding Hϵ → L r loc (R , C) is compact for ≤ r < .
We obtain the following t-Riesz formula Arguing as in [14,32,40], the function ϕ |u| possesses the following properties. (iii) ϕ |ru| = r ϕ |u| for all r ∈ R and ϕ |u(·+y)| = ϕ |u| (x + y); (iv) ϕ |u| ≥ for all u ∈ Hϵ and we have For compact supported functions in H (R , R), the following result will be very useful for some estimates below.
Proof. Assume that supp(u) ⊂ B R ( ). Since V is continuous, it is clear that Moreover, since V and A are continuous, we have and we conclude.

The modi ed problem
To study (1.1), we modify suitably the nonlinearity f so that, for ϵ > small enough, the solutions of such modi ed problem are also solutions of the original one. More precisely, we choose K > . By (f ) there exists a unique number a > verifying f (a) + a = V /K, where V is given in (V ). Hence we consider the functioñ Now we introduce the penalized nonlinearity g : where χ Λ is the characteristic function on Λ and G(x, t) := t g(x, s)ds.
From (f )-(f ), g is a Carathéodory function satisfying the following properties: is strictly increasing in t ∈ ( , +∞) and for each x ∈ Λ c , the function t → g(x,t) t is strictly increasing in ( , a). Then we consider the modi ed problem then u is a solution of problem (2.2). We observe that the weak solutions of the modi ed problem (3.2) can be found as the critical points of the C functional   Due to f is only continuous, the next results are very important because they allow us to overcome the nondi erentiability of Nϵ and the incompleteness of S + ϵ .

Lemma 3.2.
Assume that (V )-(V ) and (f )-(f ) are satis ed, then the following properties hold: (A )For any u ∈ H + ϵ , let gu : R + → R be given by gu(t) = Jϵ(tu). Then there exists a unique tu > such that g u (t) > in ( , tu) and g u (t) < in (tu , ∞); (A )There is τ > independent on u such that tu ≥ τ for all u ∈ S + ϵ . Moreover, for each compact Proof. (A ) Arguing as in [29, Lemma 3.1], it follows that gu( ) = , gu(t) > for t > small and gu(t) < for t > large. Thus, max t≥ gu(t) is achieved at a global maximum point t = tu satisfying g u (tu) = and tu u ∈ Nϵ. Now, we show that tu is unique. Arguing by contradiction, suppose that there exist t > t > such that g u (t ) = g u (t ) = . Then, for i = , , Hence, Since t > t > , we have which is a contradiction. Therefore, max t≥ gu(t) is achieved at a unique t = tu so that g u (t) = and tu u ∈ Nϵ.
From (g ), the Sobolev embeddings and < q < , it is easy to obtain which implies tu ≥ τ for some τ > . If W ⊂ S + ϵ is compact, and suppose by contradiction that there is On the other hand, let vn := tn un ∈ Nϵ, from (g ), (g ), (g ) and θ > , it yields that Thus, substituting vn := tn un and vn ϵ = tn, we may obtain as n → ∞, which yields a contradiction. This completes the proof of (A ).
(A ) We rst show that mϵ, mϵ and m − ϵ are well de ned. Indeed, by (A ), for each u ∈ H + ϵ , there is a unique mϵ(u) ∈ Nϵ. On the other hand, if u ∈ Nϵ, then u ∈ H + ϵ . Otherwise, we have |supp(u) ∩ Λϵ| = and by (g ) it follows that we know that mϵ is a bijection. Now we prove mϵ : H + ϵ → Nϵ is continuous. Let {un} ⊂ H + ϵ and u ∈ H + ϵ such that un → u in Hϵ. By (A ), there is a t > such that tn := tu n → t . Using tn un ∈ Nϵ, i.e., t n un ϵ + t n R (|x| − * |un| )|un| dx = R g(ϵx, t n |un| )t n |un| dx, ∀n ∈ N, and passing to the limit as n → ∞ in the last inequality, it follows that which implies that t u ∈ Nϵ and tu = t . This proves mϵ(un) → mϵ(u) in H + ϵ . Thus, mϵ and mϵ are continuous and (A ) is proved.
(A ) Let {un} ⊂ S + ϵ be a subsequence such that dist(un , ∂S + ϵ ) → , then for each v ∈ S + ϵ and n ∈ N, we have |un| = |un − v| a.e. in Λϵ. Thus, by (V ), (V ) and the Sobolev embedding, for any t ∈ [ , ], there exists C t > such that for all n ∈ N. From (g ), (g ) and (g ), for each t > , it follows that Therefore, On the other hand, from the de nition of mϵ and the last inequality, for all t > , we have Since t > is arbitrary, we can show that mϵ(un) ϵ → ∞ and Jϵ(mϵ(un)) → ∞ as n → ∞.
From Lemma 3.2, arguing as in [35,Corollary 10] we may obtain the following lemma. As in [35], we have the variational characterization of the in mum of Jϵ over Nϵ: Proof. Assume that {un} ⊂ Hϵ be a (PS)c sequence for Jϵ, that is, Jϵ(un) → c > and J ϵ (un) → . From (g ), (g ) and < θ < , it follows that Since K > , from the above inequalities we know that {un} is bounded in Hϵ.
The following lemma provides a range of levels in which the functional Jϵ veri es the Palais-Smale condition.
Proof. Let (un)n ⊂ Hϵ be a (PS)c for Jϵ. By Lemma 3.4, (un)n is bounded in Hϵ. Thus, up to a subsequence, un u in Hϵ and un → u in L r loc (R , C) for all ≤ r < as n → +∞.
Step 1: We show that for any given ζ > , for R large enough, Let R > such that Λϵ ⊂ B R/ ( ) and let ϕ R ∈ C ∞ (R , R) be a cut-o function such that where C > is a constant independent of R. Since the sequence (ϕ R un)n is bounded in Hϵ, we have Since ∇ Aϵ (un ϕ R ) = iun∇ϕ R + ϕ R ∇ Aϵ un, using (g ), we have By the de nition of ϕ R , the Hölder inequality and the boundedness of (un)n in Hϵ, we obtain Using the boundedness of sequence (un)n and the Sobolev embedding again, for any φ ∈ C ∞ c (R , C), we have → , as n → ∞. (3.10) By (3.8)-(3.10) and J ϵ (un) → , we have J ϵ (u) = and Step 2: Using un → u in L r loc (R , C), for all ≤ r < again, up to a subsequence, we have that |un| → |u| a.e. in R as n → +∞, then g(ϵx, |un| )|un| → g(ϵx, |u| )|u| a.e. in R as n → +∞.
By (g ) and (3.4), for any ζ > , there exists R > large enough, we have Thus, From the de nition of g, we have that Using the boundedness of (un)n in Hϵ and the diamagnetic inequality (2.1), we may assume that |∇|un|| µ and |un| ν (3.16) in the sense of measures. Moreover, by the diamagnetic inequality (2.1) and (3.4), (un)n is a tight sequence in H (R , R), thus, using the concentration-compactness principle in [37], we can nd an at most countable index I, sequences ( for any i ∈ I, where δx i is the Dirac mass at the point x i . Let us show that (x i ) i∈I ∩ Λϵ = ∅. Assume, by contradiction, that x i ∈ Λϵ for some i ∈ I. For any ρ > , we de ne ψρ( Since ∇ Aϵ (un ψρ) = iun∇ψρ + ψρ∇ Aϵ which gives a contradiction. This means that (3.15) holds.
Step 3 Since f is only assumed to be continuous, the following result is required for multiplicity result in the next section.

Corollary 3.1. The functional Ψϵ satis es the (PS)c condition on S + ϵ at any level c ∈ ( , S ).
Proof. Let {un} ⊂ S + ϵ be a (PS)c sequence for Ψϵ where c ∈ ( , S ). Then Ψϵ(un) → c and Ψ ϵ (un) * → , where · * is the norm in the dual space (Tu n S + ϵ ) * . By Lemma 3.3(B ), we know that {mϵ(un)} is a (PS)c sequence for Jϵ in Hϵ. From Lemma 3.5, we know that there exists a u ∈ S + ϵ such that, up to a subsequence, mϵ(un) → mϵ(u) in Hϵ. By Lemma 3.2(A ), we obtain un → u in S + ϵ , and the proof is completed.

Multiple solutions for the modi ed problem . The autonomous problem
Now, we study the following limit problem The solutions of problem (4.1) are the critical points of the C -functional de ned by

b )If {un} is a (PS)c sequence of Ψ , then {m(un)} is a (PS)c sequence of I . If {un} ⊂ N is a bounded (PS)c sequence of I , then {m − (un)} is a (PS)c sequence of Ψ ; (b )u is a critical point of Ψ if and only if m(u) is a critical point of I . Moreover, the corresponding critical values coincide and
Similar to the previous argument, we also have the following variational characterization of the in mum of I over N : thus, u is a ground state solution. From the assumption of f , u ≥ . Moreover, using the standard argument, we may prove that u(x) > for x ∈ R . The proof is complete.
Note that, arguing as in [26, Proposition 3.3, Proposition 3.4 and Lemma 3.11], the ground state solution of problem decays exponentially at in nity with its gradient, and is C (R , R) ∩ L ∞ (R , R). This result is very important for the proof of Lemma 4.6 later.  Then tϵ satis es where we use supp(η) ⊂ Λ and the de nition of g(x, t). Moreover, combining the facts that η = in B ρ/ , u is a positive continuous function and hypothesis (f ), we have for all < ϵ < and where γ = min{ω(z) : |z| ≤ ρ/ }. If tϵ → +∞ as ϵ → , by (f ), we deduce that R (|x| − * |ωϵ| )|ωϵ| dx → +∞ which contradicts (4.5). Therefore, up to a subsequence, we may assume that tϵ → t ≥ as ϵ → . If tϵ → , using the fact that f is increasing, the Lebesgue dominated convergence theorem and relation (4.5), we obtain which contradicts (4.3). Thus, we have t > and Since ω ∈ N V , we obtain that t = and so, using the Lebesgue dominated convergence theorem, we get Hence lim ϵ→ Jϵ(tϵ ωϵ) = I V (u) = c V .

The technical results
By the Ljusternik-Schnirelmann category theory, in this subsection we prove a multiplicity result for the modi ed problem (3.2). We rst provide some useful preliminary results.
By construction, Φϵ(y) has compact support for any y ∈ M. Moreover, arguing as in Lemma 4.1, the energy of above function has the following behavior as ϵ → + .
Lemma 4.7. The limit lim ϵ→ + Jϵ(Φϵ(y)) = c V holds uniformly in y ∈ M. Now we de ne the barycenter map. Let ρ > be such that M δ ⊂ Bρ and consider Υ : R → R de ned by setting The barycenter map βϵ : Nϵ → R is de ned by Lemma 4.8. The limit lim ϵ→ + βϵ(Φϵ(y)) = y holds uniformly in y ∈ M.
Proof. Assume by contradiction that there exists κ > , (yn) ⊂ M and ϵn → such that |βϵ n (Φϵ n (yn)) − yn| ≥ κ. , then (tn) is also bounded and so, up to a subsequence, we may assume that tn → t ≥ . We claim that t > . Indeed, if t = , then, since (|vn|)n is bounded, we haveṽn → in H (R , R), that is I (ṽn) → , which contradicts c V > . Thus, up to a subsequence, we may assume thatṽn ṽ := t v ≠ in H (R , R), and, by Lemma 4.5, we can deduce thatṽn →ṽ in H (R , R), which gives |vn| → v in H (R , R). Now we show the nal part, namely that {yn} has a subsequence such that yn → y ∈ M. Assume by contradiction that {yn} is not bounded and so, up to a subsequence, |yn| → +∞ as n → +∞. Choose R > such that Λ ⊂ B R ( ). Then for n large enough, we have |yn| > R, and, for any x ∈ B R/ϵn ( ), Since un ∈ Nϵ n , using (V ) and the diamagnetic inequality (2.1), we get that (4.8) Since |vn| → v in H (R , R) andf (t) ≤ V /K, we can see that (4.8) yields that is |vn| → in H (R , R), which contradicts to v ≢ . Therefore, we may assume that yn → y ∈ R . Assume by contradiction that y ∉ Λ. Then there exists r > such that for every n large enough we have that |yn − y | < r and B r (y ) ⊂ Λ c . Then, if x ∈ B r/ϵn ( ), we have that |ϵn x + yn − y | < r so that ϵn x + yn ∈ Λ c and so, arguing as before, we reach a contradiction. Thus, To prove that V(y ) = V , we suppose by contradiction that V(y ) > V . Using the Fatou's lemma, the change of variable z = x +ỹn and max t≥ Jϵ n (tun) = Jϵ n (un), we obtain Let nowÑ Fixed y ∈ M, since, by Lemma 4.7, |Jϵ(Φϵ(y)) − c V | → as ϵ → + , we get thatÑϵ ≠ ∅ for any ϵ > small enough.
The relation betweenÑϵ and the barycenter map is as follows. Therefore, it is enough to prove that there exists (yn) ⊂ M δ such that lim n |βϵ n (un) − yn| = .
By the diamagnetic inequality (2.1), we can see that I (t|un|) ≤ Jϵ n (tun) for any t ≥ . Therefore, recalling that {un} ⊂Ñϵ n ⊂ Nϵ n , we can deduce that Since, up to a subsequence, |un|(· +ỹn) converges strongly in H (R , R) and ϵn z + yn → y ∈ M for any z ∈ R , we conclude.
. Multiplicity of solutions for problem (3.2) Finally, we present a relation between the topology of M and the number of nontrivial solutions of the modied problem (3.2). Therefore, there is a numberε > such that the setS + ϵ : Here h is given in the de nition ofÑϵ. Given δ > , by Lemma 4.7, Lemma 3.2(A ), Lemma 4.8, and Lemma 4.9, we can ndε δ > such that for any ϵ ∈ ( ,ε δ ), the following diagram (4.10) By Corollary 3.1 and the abstract category theorem [35], Ψϵ has at least cat πϵ(M) (πϵ(M)) critical points on S + ϵ . Therefore, from Lemma 3.3(B ) and (4.10), we have that Jϵ has at least cat M δ (M) critical points inÑϵ which implies that problem (3.2) has at least cat M δ (M) solutions.

Proof of Theorem 1.1
In this section we shall show that the solutions uϵ obtained in Theorem 4.1 satisfy |uϵ(x)| ≤ a for x ∈ Λ c ϵ for ϵ small and prove the main result of this paper.
Arguing as in [29] or [41], the following uniform result holds. We argue by contradiction and assume that there is a sequence ϵn → such that for every n there exists un ∈Ñϵ n which satis es J ϵn (un) = and un L ∞ (Λ c ϵn ) > a. (5.2) As in Lemma 5.1, we have that Jϵ n (un) → c V , and therefore we can use Proposition 4.1 to obtain a sequence (ỹn) ⊂ R such that yn := ϵnỹn → y for some y ∈ M. Then, we can nd r > , such that Br(yn) ⊂ Λ, and so B r/ϵn (ỹn) ⊂ Λϵ n for all n large enough. Using Lemma 5.1, there exists R > such that |vn| ≤ a in B c R ( ) and n large enough, where vn = un(· +ỹn). Hence |un| ≤ a in B c R (ỹn) and n large enough. Moreover, if n is so large that r/ϵn > R, then Λ c ϵn ⊂ B c r/ϵn (ỹn) ⊂ B c R (ỹn), which gives |un| ≤ a for any x ∈ Λ c ϵn . This contradicts (5.2) and proves the claim. Let now ϵ δ := min{ε δ ,ε δ }, whereε δ > is given by Theorem 4.1. Then we have cat M δ (M) nontrivial solutions to problem (3.2). If uϵ ∈Ñϵ is one of these solutions, then, by (5.1) and the de nition of g, we conclude that uϵ is also a solution to problem (2.2). Finally, we study the behavior of the maximum points of |ûϵ|, whereûϵ(x) := uϵ(x/ϵ) is a solution to problem (1.1), as ϵ → + . Take ϵn → + and the sequence (un) where each un is a solution of (3.2) for ϵ = ϵn. From the de nition of g, there exists γ ∈ ( , a) such that g(ϵx, t )t ≤ V K t , for all x ∈ R , |t| ≤ γ.
Arguing as above we can take R > such that, for n large enough, Up to a subsequence, we may also assume that for n large enough un L ∞ (B R (ỹn)) ≥ γ. (5.4) Indeed, if (5.4) does not hold, up to a subsequence, if necessary, we have un ∞ < γ. Thus, since J ϵn (uϵ n ) = , using (g ) and the diamagnetic inequality (2.1) that R (|∇|un|| + V |un| )dx ≤ R g(ϵn x, |un| )|un| dx ≤ V K R |un| dx and, being K > , un = , which is a contradiction. Taking into account (5.3) and (5.4), we can infer that the global maximum points pn of |uϵ n | belongs to B R (ỹn), that is pn = qn +ỹn for some qn ∈ B R . Recalling that the associated solution of problem (1.1) isûn(x) = un(x/ϵn), we can see that a maximum point ηϵ n of |ûn| is ηϵ n = ϵnỹn + ϵn qn. Since qn ∈ B R , ϵnỹn → y and V(y ) = V , the continuity of V allows to conclude that lim n V(ηϵ n ) = V .