Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{2}$$\end{document}

In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation -(∇+iA(x))2u+(λV(x)+Z(x))u=f(|u|2)uinR2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -(\nabla +\mathrm{i} A(x))^2 u+(\lambda V(x)+Z(x))u=f(\vert u\vert ^{2})u\quad \text {in}\, \,{\mathbb {R}}^{2}, \end{aligned}$$\end{document}where λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document}, f(t) is a continuous function with exponential critical growth, the magnetic potential A:R2→R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$$\end{document} is in Lloc2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}_{loc}({\mathbb {R}}^2)$$\end{document} and the potentials V, Z:R2→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z:{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}$$\end{document} are continuous functions verifying some natural conditions. We show that if the zero set of the potential V has several isolated connected components Ω1,…,Ωk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{1}, \ldots , \Omega _{k}$$\end{document} such that the interior of Ωj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{j}$$\end{document} is non-empty and ∂Ωj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _{j}$$\end{document} is smooth, then for λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} large enough, the equation has at least 2k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{k}-1$$\end{document} multi-bump solutions.


Introduction and main results
This paper is devoted to the qualitative analysis of solutions for the nonlinear magnetic Schrödinger equation in R 2 . We are concerned with the existence and multiplicity of multi-bump solutions if the reaction has an exponential critical behavior. In the first part of this section, we recall some significant historical moments related to the development of the Schrödinger theory. The main result and an associated multiplicity property are described in the second part of the present section.
the future behaviour of a dynamical system. It is striking to point out that talking about his celebrating equation, Erwin Schrödinger said: "I don't like it, and I'm sorry I ever had anything to do with it". The linear Schrödinger equation is a central tool of quantum mechanics, which provides a thorough description of a particle in a non-relativistic setting. Schrödinger's linear equation is where u is the Schrödinger wave function, m is the mass of the particle, denotes Planck's renormalized constant, E is the energy, and V stands for the potential energy.
Schrödinger also established the classical derivation of his equation, based upon the analogy between mechanics and optics, and closer to de Broglie's ideas. He developed a perturbation method, inspired by the work of Lord Rayleigh in acoustics, proved the equivalence between his wave mechanics and Heisenberg's matrix, and introduced the time dependent Schrödinger's equation In physical problems, a cubic nonlinearity corresponding to p = 3 in Eq. (1.1) is common; in this case problem (1.1) is called the Gross-Pitaevskii equation. In the study of Eq. (1.1), Floer and Weinstein [24] and Oh [37] supposed that the potential V is bounded and possesses a non-degenerate critical point at x = 0. More precisely, it is assumed that V belongs to the class (V a ) (for some real number a) introduced in Kato [30]. Taking γ > 0 and > 0 sufficiently small and using a Lyapunov-Schmidt type reduction, Oh [37] proved the existence of bound state solutions of problem (1.1), that is, a solution of the form u(x, t) = e −i Et/ u(x) . (1.2) Using the Ansatz (1.2), we reduce the nonlinear Schrödinger equation (1.1) to the semilinear elliptic equation The change of variable y = −1 x (and replacing y by x) yields Let us also recall that in his 1928 pioneering paper, Gamow [25] proved the tunneling effect, which lead to the construction of the electronic microscope and the correct study of the alpha radioactivity. The notion of "solution" used by him was not explicitly mentioned in the paper but it is coherent with the notion of weak solution introduced several years later by other authors such as J. Leray, L. Sobolev and L. Schwartz. Most of the study developed by Gamow was concerned with the bound states u(x, t) defined in (1.2), where u solves the stationary equation for a given potential V (x). Gamow was particularly interested in the Coulomb potential but he also proposed to replace the resulting potential by a simple potential that keeps the main properties of the original one. In this way, if is a subdomain of R N , Gamow proposed to use the finite well potential It seems that the first reference dealing with the limit case, the so-called infinite well potential, was the book by the 1977 Nobel Prize Mott [36]. The more singular case in which V 0 is the Dirac mass δ 0 is related with the so-called Quantum Dots, see Joglekar [29].
In contrast with classical mechanics, in quantum mechanics the incertitude appears (the Heisenberg principle). For instance, for a free particle (i.e. with V (x) ≡ 0), in nonrelativistic quantum mechanics, if the wave function u(·, t) at time t = 0 vanishes outside some compact region then at an arbitrarily short time later the wave function is nonzero arbitrarily far away from the original region . Thus, the wave function instantaneously spreads to infinity and the probability of finding the particle arbitrarily far away from the initial region is nonzero for all t > 0. We refer to Díaz [20] for more details. Finally, we point out that sublinear Schrödinger equations with lack of compactness and indefinite potentials have been studied by Bahrouni, Ounaies and Rȃdulescu [10,11].

Main results
Consider the following nonlinear Schrödinger equation where λ > 0 is a parameter, V, Z , f are continuous functions verifying some assumptions, has been studied by many researchers. In [21], Ding and Tanaka studied problem (1.4) assuming f (t) = |t| q−1 t. In this mentioned paper, the authors established the existence of multi-bump positive solutions for the problem where 2 < p < 2N /(N − 2) for N ≥ 3 and 2 < p < ∞ for N = 1, 2. The authors showed that problem (1.5) has at least 2 k − 1 multi-bump solutions for λ large enough. These solutions have the following properties: for each non-empty subset ⊂ {1, 2, . . . , k} and > 0 fixed, there is λ * > 0 such that problem (1.5) possesses a solution u λ for all λ ≥ λ * = λ * ( ), satisfying: where = j∈ j and c j is the minimax level of the energy functional related to In [2], using variational methods, Alves et al. considered the existence of multibump positive solutions for the following problem with critical growth In [2], due to the critical growth of the nonlinearity in R N , the method applied in [21] does not hold. In [8], using a new approach, Alves et al. established the same results for the following equation where f is continuous with exponential critical growth. Due to the exponential critical growth of the nonlinearity in R 2 , some estimates in [8] are completely different from the case N ≥ 3. For the further research about the nonlinear Schrödinger equation with the deepening potential well, we refer to [1,[4][5][6][7]9,12,27,34] and their references. In recent years, the nonlinear magnetic Schrödinger equation has also received considerable attention. This class of problems has some relevant physical applications, such as nonlinear optics and plasma physics. The function u(x, t) takes on complex values, is the Planck constant, i is the imaginary unit, A : R 2 → R 2 is the magnetic potential. When one looks for standing wave solutions u(x, t) := e −i Et/ u(x), with E ∈ R, of Eq. (1.8), the problem can be reduced by (1.9) As far as we know, the first result seems to be established in [23], where the existence of standing waves to problem (1.9) has been obtained for > 0 fixed and for special classes of magnetic fields. In this way, the authors obtained the existence of solutions by solving an appropriate minimization problem for the corresponding energy functional in the cases N = 2 and N = 3. After that, Kurata [31] proved that the problem has a least energy solution for any > 0 when a technical condition relating V (x) and A(x) is assumed. Under this technical condition, Kurata proved that the associated energy functional satisfies the Palais-Smale condition at any level. In [3], by combining a local assumption on V , the penalization techniques of del Pino and Felmer [19] and the Ljusternik-Schnirelmann theory, Alves et al. obtained the multiple solutions. We would like to refer to [16][17][18]22,28,35] for other results related with the problem (1.9).
Recently, there are many works concerning the following magnetic Schrödinger equation with deepening potential well (1.10) In particular, Tang [39] considered multi-bump solutions of problem (1.10) with critical frequency in which Z (x) ≡ 0 and f satisfies subcritical growth. Liang and Shi [33] considered multi-bump solutions of problem (1.10) with critical nonlinearity for the case N ≥ 3. It is quite natural to consider multi-bump solutions for the problem when the nonlinearity satisfies the exponential critical growth in N = 2.
To the best of our knowledge, this problem has not been considered. Motivated by [3,8,33], the main goal of the present paper is to prove the existence of multi-bump solutions for problem (1.10), considering a class of nonlinearity with exponential critical growth in R 2 . Because the nonlinearity has exponential critical growth in R 2 , some properties that are valid for N ≥ 3, do not necessarily hold for the class of problems studied in this paper. Therefore, we need to take different approaches in some estimates. On the other hand, as we will see later, due to the presence of the magnetic field A(x), problem (1.10) cannot be changed into a pure real-valued problem, hence we should deal with a complex-valued directly, which causes several new difficulties in employing the methods in dealing with our problem. Our problem is more complicated than the pattern studied in [8] and we need additional technical estimates. We now present the general assumptions used in the statement of the main result of this paper.
We assume that the reaction f is a continuous function satisfying the following conditions.
There exist constants p > 2 and C p > 0 such that The main result in this paper is stated below. . . , k}, there exists λ * such that for all λ ≥ λ * , problem (1.10) has a nontrivial solution u λ . Moreover, the family {u λ } λ≥λ * has the following properties: for any sequence λ n → ∞, we can extract a subsequence λ n i such that u λ n i converges strongly in H 1 A (R 2 , C) to a function u which satisfies u(x) = 0 for x / ∈ and the restriction u| j is a least energy solution of

Corollary 1.2. Under the assumptions of Theorem
The paper is organized as follows. In Sect. 2 we introduce the functional setting and we give some preliminary results. In Sect. 3, we study the modified problem. We prove the Palais-Smale condition for the modified energy functional for λ large and study L ∞ -estimates for the solution and the behavior of (P S) ∞ sequences. In Sect. 4, we adapt the deformation flow method in order to establish the existence of a special critical point, which is crucial for showing the existence of multi-bump solutions for λ large enough and hence to prove Theorem 1.1. We refer to the recent monograph by Papageorgiou, Rȃdulescu and Repovš [38] for some of the abstract methods used in this paper.

Notation
• C, C 1 , C 2 , . . . denote positive constants whose exact values are inessential and can change from line to line; • B R (y) denotes the open disk centered at y ∈ R 2 with radius R > 0 and B c R (y) denotes the complement of B R (y) in R 2 ; • · , · q , and · L ∞ ( ) denote the usual norms of the spaces

Abstract setting and preliminary results
In this section, we outline the variational framework for problem (1.10) and give some auxiliary properties.
For u : R 2 → C, let us denote by and 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. The spaces H 1 we have the following diamagnetic inequality (see e.g. [32,Theorem 7.21]): with the norm For λ ≥ 1, it is easy to see that (E λ (R 2 , C), · λ ) is a Hilbert space and We recall that u ∈ E λ (R 2 , C) is a weak solution of (1.10), if The weak solutions of problem (1.10) are the critical points of I λ : E λ (R 2 , C) → R given by where u 2 2,K = K |u| 2 dx. The following property is an immediate consequence of the above consideration.
The below estimates involving f are the key points in this paper. By ( f 1 ) and ( f 2 ), fixed q > 2, for any ζ > 0 and α > 4π , there exists a constant C > 0 depending on q, α, ζ , such that and, using ( f 3 ), we have Moreover, by (2.2) and (2.3), and Now we recall a version of the Trudinger-Moser inequality in the whole space R 2 due to Cao [15] (see also [13], Lemma 2.3) for functions belonging to H 1 (R 2 , R).
Moreover, if ∇u 2 2 ≤ 1, u 2 ≤ M < +∞, and 0 < α < 4π , then there exists a positive constant C(M, α), which depends only on M and α, such that To finish this section, in what follows, for each j ∈ {1, 2, . . . , k}, we fix a bounded open subset j with smooth boundary such that

An auxiliary problem
Since R 2 is unbounded, we know that the Sobolev embeddings are not compact, as so I λ cannot verify the Palais-Smale condition. In order to overcome this difficulty, we adapt an argument of the penalization method introduced by del Pino and Felmer [19] and Ding and Tanaka [21]. Let ν 0 > 0 be a constant given in Lemma 2.2, κ > θ θ−2 > 1 and a > 0 verifying f (a) = ν 0 κ andf , F : R → R given bỹ and Note that From now on, we fix a non-empty subset ⊂ {1, . . . , k} and It follows from (3.1) that g satisfies the following inequality Standard arguments show that λ ∈ C 1 (E λ (R 2 , C), R) and its critical points are weak solutions of Our aim is to obtain nontrivial solutions of (3.5) which are solutions of the original problem (1.10). More precisely, if u λ is a nontrivial solution of (3.5) verifying |u λ (x)| 2 ≤ a in R 2 \ , then it is a nontrivial solution to (1.10).

The Palais-Smale condition and consequences
We start this subsection studying the boundedness of the Palais-Smale sequence related to λ , that is, a sequence (u n ) ⊂ E λ (R 2 , C) verifying λ (u n ) → c and λ (u n ) → 0 for some c ∈ R(shortly (u n ) is a (P S) c sequence).
For each fixed j ∈ , let us denote by c j the minimax level of the functional I j : H 0,1 A ( j ) → R given by and It is well known that the critical points of I j are weak solutions of the following problem In the next lemma, we denote by S the following real number Proof. For each j ∈ {1, . . . , k}, we may choose a function ϕ j ∈ H 0,1 A ( j , C) such that On the other hand, by ( f 4 ) we have Since δ 0 may be chosen close to 1, the last inequality implies that This completes the proof of the lemma. Thus, up to a subsequence, u n u in E λ (R 2 , C) and u n → u in L q loc (R 2 , C) for all q ≥ 1 as n → +∞. Moreover, by (3.6) and (2.2), fixed q > 2, for any ζ > 0 and α > 4π , there exists a constant C > 0, which depends on q, α, ζ , such that for any φ ∈ E λ (R 2 , C), Arguing as in [18, Lemma 2.5], we have Thus, u is a critical point of λ . Now, we take R > 0 such that ⊂ B R 2 (0). Let φ R ∈ C ∞ (R 2 , R) be a cut-off function such that where C > 0 is a constant independent of R. By a direct computation, one has Therefore, Notice that Using the Hölder inequality and (3.7) we obtain lim sup n→∞ Re( Moreover, we have which implies that for any ζ > 0, there exists R * > 0 large, if R > R * , one has lim sup Similarity, by (3.6) and (2.2), fixed q > 2, for any ζ > 0 and α > 4π , there exists a constant C > 0, which depends on q, α, ζ , such that g(x, |u n | 2 )|u n | 2 ≤ ζ |u n | 2 + C|u n | q (e α|u n | 2 − 1). (3.9) Since u n → u in L r loc (R 2 , C), for all r ≥ 1, up to a subsequence, we have that |u n | → |u| a.e. in R 2 as n → +∞.
Finally, since λ (u) = 0, we have Thus, the sequence (u n ) strong converges to u in E λ (R 2 , C).
Our next step is to study the behavior of a (P S) ∞,c sequence, that is, a sequence (u n ) ⊂ H 1 A (R 2 , C) satisfying u n ∈ E λ n (R 2 , C) and λ n → ∞, Moreover, and u| j is a solution of (3.6), for ∀ j ∈ ; (ii) u n − u λ n → 0; (iii) u n also satisfies Proof. As in the proof of Proposition 3.3, it is easy to check that lim sup n→∞ u n 2 λ n < 1.

Thus (u n ) is bounded in H 1
A (R 2 , C) and we may assume that for some u ∈ H 1 A (R 2 , C), up to a subsequence, if necessary u n u weakly in H 1 A (R 2 , C), u n → u strongly in L r loc (R 2 , C), ∀r ≥ 1, |u n | → |u| a.e. in R 2 . (3.10) To show (i), we fix the set C m = {x ∈ R 2 : V (x) ≥ 1 m }. Then, for n large The last inequality together with Fatou's lemma imply Therefore, u(x) = 0 on +∞ m=1 C m = R 2 \ , from which we can assert that u| j ∈ H 0,1 A ( j , C) for any j ∈ {1, 2, . . . , k}.
For each j ∈ {1, . . . , k}\ , setting ϕ = u| j in (3.11), we have By Lemma 2.2 and the fact thatf (t 2 )t 2 ≤ ν 0 κ t 2 for all t ∈ R, it yields Thus u = 0 in j for j ∈ {1, 2, . . . , k}\ , it means that (i) holds. For (ii), using the similar arguments in the proof of Proposition 3.3, for each ζ > 0, there exists R > 0 such that Using the same arguments as in the proof of Proposition 3.3 and (i), the above inequality implies that Now, by (i) again, we have Thus, the sequence (u n ) strong converges to u in E λ n (R 2 , C) and (ii) holds.
To prove (iii), notice that from (i) and (ii), Moreover, from (i) and (ii), it is also easy to obtain that as n → ∞ Therefore, the proof is complete.
Proof. Let (λ n ) be a sequence with λ n → ∞ and define u n (x) = u λ n (x). For any For each n ∈ N and L > 0, we consider the functions where β > 1 will be determined later. By straightforward computations, we have and Taking the real part of ∇ A u n ∇ A z L ,n and using the diamagnetic inequality (2.1), we obtain Taking z L ,n as the test function, we have g(x, |u n | 2 )u n z L ,n dx.
Proof. We use notation B r (x) = {y ∈ R 2 : |x − y| < r }. Since u λ ∈ E λ (R 2 , C) is a critical point of λ (u), that is, u λ satisfies the following equation

By Kato's inequality
there holds Using Proposition 3.5 and the subsolution estimate (see [26] By Proposition 3.4, for any sequence λ n → ∞, we can extract a subsequence λ n i such that In particular, Since λ n → ∞ is arbitrary, we have The proof is now complete.

The existence of multi-bump positive solutions
In this section, for each j ∈ , we denote by λ, j : H 1 A ( j , C) → R the functional given by It is easy to check that the functional λ, j satisfies the mountain pass geometry. In what follows, we denote by c λ, j the minimax level related to the above functional defined by Therefore, there exist (u n ) ⊂ H 0,1 A ( j , C) and (u λ,n ) ⊂ H 1 A ( j , C) verifying I j (u n ) → c j and I j (u n ) → 0, and λ, j (u λ,n ) → c λ, j and λ, j (u λ,n ) → 0.
and these inequalities imply that I j and λ, j satisfy the (P S) c j and (P S) c λ, j conditions, respectively. Therefore, it is easy to prove that there exist two nontrivial functions w j ∈ H 0,1 A ( j ) and w λ, j ∈ H 1 A ( j ) verifying I j (w λ, j ) = c j and I j (w λ, j ) = 0, and λ, j (w λ, j ) = c λ, j and λ, j (w λ, j ) = 0.
Moreover, we have the following lemma.
Proof. From ( f 4 ), it is easy to prove that c λ, j > 0 and c j > 0 for any j ∈ and λ ≥ 1. Now for any u ∈ H 0,1 . Thus, we have j ⊂ λ, j and Thus (i) holds. The proof of (ii) and (iii) is standard by using the monotonicity of the term f (t) with respect to t for t > 0. Now we prove (iv). Using Proposition 3.4, we may extract a subsequence λ n → ∞ such that is a solution of (3.6) and By the definition of c j , we have lim sup Together with (i), we get (iv).

A special critical value of λ
In what follows, let us fix R > 1 such that From the definition of c j , it is easy to check that   1 , s 2 , . . . , s l )).
The proof is thus completed.
Proof. (i) For all λ ≥ 1 and for each j, we have 0 < c λ, j ≤ c j . Using the same arguments in the proof of Proposition 3.4, we can prove that c λ, j → c j as λ → ∞ and thus, from Proposition 4.3, b λ, → c as λ → ∞.
(ii) Using the fact that λ verifies that Palais-Smale condition, we can use well known arguments involving deformation lemma [40] to conclude that b λ, is a critical level to λ for large λ.