An Existence Result for a Class of Magnetic Problems in Exterior Domains

In this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation (P)(-i∇+A(x))2u+u=|u|p-2u,inΩ,u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = |u|^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$\end{document}where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document}, Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N$$\end{document} is an exterior domain, p∈(2,2∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (2, 2^*)$$\end{document} with 2∗=2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^*=\frac{2N}{N-2}$$\end{document}, and A:RN→RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$$\end{document} is a continuous vector potential verifying A(x)→0as|x|→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .$$\end{document}


Introduction
In this paper we investigate the existence of solutions for the following magnetic semilinear Schrödinger equation where N ≥ 3, p ∈ (2, 2 * ), 2 * := 2N N −2 is the critical Sobolev exponent, Ω ⊂ R N is an exterior domain, i.e. Ω is an unbounded domain with smooth boundary ∂Ω = ∅ such that R N \Ω is bounded, and A ∈ C(R N , R N ) satisfies A(x) → 0 as |x| → ∞. (A) During the past years there has been a considerable interest in the existence of solutions for elliptic equations in exterior domains, more precisely, for problems of the type −Δu + u = f (u), in Ω, u ∈ H 1 0 (Ω), (1.1) where f is a continuous function satisfying some technical conditions. The main difficulty in dealing with (1.1) is the lack of compactness of the Sobolev embedding due to the unboundedness of the domain. In order to overcome this difficulty, in some papers, authors assumed certain type of symmetry on Ω; see for instance [9], [23] and [25].
In that article, they proved that (1.2) does not have a ground state solution and this fact yields a series of difficulties. The key idea exploited by the authors was to analyze the behavior of Palais-Smale sequences, obtaining a precise estimate of the energy levels where the Palais-Smale condition fails. The authors proved that if p = 1 + 8 N for N = 5, 6, 7 or p < 2(N −1) N −2 for N = 3, 4, • there exists λ * > 0 such that, for every λ ∈ (0, λ * ), (1.2) has at least one positive solution, • for every λ there exists a ρ = ρ(λ) such that if R N \Ω ⊂ B(x 0 , ρ), with x 0 ∈ R N \Ω, (1.2) has at least one positive solution.
Later, existence results were obtained for more general problems where f is a continuous function satisfying lim |x|→+∞ f (x, t) = f ∞ (t), for all t ∈ R.
In [9], Bahri and Lions studied (1. as |x| → +∞. Using topological arguments, they showed that (1.3) has a solution when Ω is an arbitrary exterior domain, for all λ > 0. In the autonomous case, using the technique introduced in [9], Li and Zheng [28] proved that (1.3) possesses at least one positive solution, with f asymptotically linear satisfying some assumptions, in particular, a property of convexity (see also [21]). Whereas in [29], Maia and Pellacci established an existence result without the hypothesis of convexity.
In the above-mentioned papers, a key point to prove the results of existence is the uniqueness, up to a translation, of the positive solution for the "equation at infinity" associated with (1.3) given by However, when the exterior domain is the exterior of a ball, more precisely Ω = R N \B(0, R) for some R > 0, it is possible to explore some groups of rotation in order to get multiple solutions without using the uniqueness of solutions of the limit problem when R is large enough, see for example Cao and Noussair [16] and Clapp, Maia and Pellacci [22]. In [2], Alves and de Freitas studied the existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Finally, we mention a recent paper due to Alves, Molica Bisci and Torres Ledesma [5], in which a fractional elliptic equation with Dirichlet-type condition set in an exterior domain is considered.
In the last years time-independent magnetic Schrödinger equations in bounded domains or in whole of R N have received a special attention. A basic motivation to study these equations stems from the search of standing wave solutions for the time-dependent nonlinear Schrödinger equation of the type  [3,4], Ambrosio [6,7], Arioli and Szulkin [8], Barile [10,11], Chabrowski and Szulkin [17], Cingolani [18], Cingolani and Clapp [19], Cingolani, Jeanjean and Secchi [20], Ding and Liu [24], Esteban and Lions [26] and the references therein. After a careful bibliography review, we did not find any paper concerned with magnetic semilinear Schrödinger equations in exterior domains. Motivated by this fact and the above-mentioned papers, the aim of this paper is to give a first existence result for (P ). We emphasize that the main difficulty in dealing with this type of problem is related to the uniqueness, up to a translation, of the solution for the limit problem. Recently, in a very interesting paper due to Bonheure, Nys and Van Schaftingen [14], we found a partial answer for this question when the magnetic field A satisfies some technical conditions; see [14,Theorem 1].
The main result of this paper can be stated as follows: , ρ < ρ 0 and A ∞ ≤ , then problem (P ) has at least one weak solution.
The proof of Theorem 1.1 will be done done via variational methods inspired by [2] and [12]. However, with respect to [2,12], a more careful analysis will be needed and some refined estimates will be given. The diamagnetic inequality in [26] will play a fundamental role.
We point out that Theorem 1.1 complements the study in magnetic semilinear Schrödinger equations, in the sense that we obtain an existence result for a magnetic Schrödinger equation in an exterior domain.
The paper is organized as follows. In Sect. 2, we introduce suitable function spaces and collect some useful results concerning the limit problem that we will 526 C. O. Alves et al. Vol. 89 (2021) work with. In Sect. 3, we establish a compactness result in the spirit of [12] for the energy functional associated with problem (P ). In Sect. 4, we show some technical estimates that will be used in Sect. 5, where Theorem 1.1 is proved. Notations: In this paper, we use the following notations: • For q ∈ (2, 2 * ), we define q as the conjugate exponent of q, that is, q := q q−1 . • The usual norm of the Lebesgue spaces L t (Ω) for t ∈ [1, ∞], will be denoted by | . | t , and the norm of the Sobolev space H 1 0 (Ω), by . ; • C denotes (possibly different) any positive constant.

Preliminary Results and the Limit Problem
In what follows, we denote by H 1 A (Ω, C) the Hilbert space obtained by the closure of C ∞ 0 (Ω, C) under the scalar product where (w) denotes the real part of w ∈ C, w is its complex conjugate, The norm induced by this inner product is given by We also consider the Hilbert space Then we can define the norm Hence, if u ∈ H 1 A (R N , C), then |u| ∈ H 1 (R N , R). Furthermore, as a consequence of the diamagnetic inequality, we have that the embedding Magnetic Problem on Exterior Domains 527 is continuous for any s ∈ [2, 2 * ].

Limit Problem
In this subsection, we consider the scalar limit problem associated with (P ), namely where p ∈ (2, 2 * ). The reader is invited to see that u = u 1 + ıu 2 , with u 1 and u 2 real valued, is a solution of (P 0 ) if and only if u 1 and u 2 solve the following elliptic system which is a system of the gradient type. Note that, the solutions of (P 0 ) are critical points of the functional If c 0 denotes the mountain pass level of I 0 and N 0 is the Nehari manifold defined as N 0 := {u ∈ H 1 0 (R N , C)\{0} : I 0 (u)u = 0}, it is well-known (see [30]) that from where it follows that c 0 is the least energy of (P 0 ). We recall that u ∈ H 1 0 (R N , C) is a least energy solution of (P 0 ) if I 0 (u) = c * 0 and I 0 (u) = 0, and c * 0 is called the least energy of (P 0 ).
Lemma 2.1. The following fact holds: u is a least energy solution of (P 0 ) if, and Moreover, (P 0 ) and (P ∞ ) have the same least energy.
Proof. The proof can be done as in [24, Lemma 2.5].

Lemma 2.2. The following facts hold:
(1) c 0 > 0 is the least energy of (P ∞ ); (2) N 0 = ∅; (3) c 0 is attained, and the set Let I ∞ : H 1 (R N , R) → R be the energy functional given by If c ∞ denotes the mountain pass level of I ∞ and N ∞ is the Nehari manifold given by then (see [30]) Let ϕ ∈ H 1 (R N , R) be a positive ground state solution of (P ∞ ), that is, The function ϕ can be chosen radial and decreasing with respect to |x|; see [13]. An immediate consequence of Lemmas 2.1 and 2.2, we have the equality c 0 = c ∞ , and so, ϕ is also a ground state solution for (P 0 ).
Before concluding this section, we state an important result that is a particular case of a result due to Bonheure, Nys and Van Schaftingen [14, Theorem 1], which will be crucial in our approach.

A Compactness Result for Energy Functional
In this section, we study some compactness property of the energy functional I A : In the sequel, we denote by c A the mountain pass level of I that satisfies the equality below where N A is the Nehari manifold of I A given by and so, problem (P ) has no ground state solution.
Proof. By using the diamagnetic inequality (2.1), Recalling that ϕ satisfies (2.4) and that c 0 = c ∞ , we have that Let (y n ) ⊂ Ω be a sequence such that |y n | → +∞, and ρ be the smallest positive number satisfying Now, we consider the sequence and fix t n > 0 such that t n ψ n ∈ N A . By making the change of variable z = x − y n , we deduce Since |y n | → +∞ as n → +∞, it is easy to check that As the dominated convergence theorem ensures that or equivalently, Alves et al. Vol. 89 (2021) Recalling that ζ(x) = 0 for x ∈ R N \Ω, by the previous discussion, On the other hand, for each j ∈ {1, .., N }, and so, By the previous analysis together with the fact that ϕ ∈ N 0 , using translation invariance it is not difficult to prove that t n → 1. Thus, by definition of c A , (3.3) and (3.4), we get From (3.2) and (3.5), By (2.1), (3.6), and recalling that c 0 = c ∞ , we deduce that the function w = |v 0 | ∈ H 1 0 (R N , R) is a ground state solution of (P ∞ ), that is, Since w ≥ 0 in R N and w = 0, the strong maximum principle ensures that w(x) > 0 for all Vol. 89 (2021) Magnetic Problem on Exterior Domains 531

A Compactness Lemma
In this section, we prove a compactness result involving the energy functional I A associated with (P ). In order to do this, we need to consider the energy functional (2.3). With the above notations, we are able to prove the following compactness result.
Then, up to a subsequence, there exists a weak solution u 0 ∈ H 1 A (Ω, C) of (P ) such that Proof. We proceed by steps.
Step 1. The sequence (u n ) is bounded in H 1 A (Ω, C). By (3.7), Choosing ψ = u n in (3.8), we obtain which combined with (3.9) and (3.10) gives  Consequently, up to a subsequence, there exists u 0 ∈ H 1 A (Ω, C) such that u n u 0 in H 1 A (Ω, C), u n → u 0 in L p loc (Ω, C) for p ∈ [2, 2 * ), u n (x) → u 0 (x) a.e. in Ω. (3.12) We claim that u 0 is a weak solution of (P ). In fact, for an arbitrary function ψ ∈ H 1 A (Ω, C), the limit From the boundedness of (u n ) in H 1 A (Ω, C) and Sobolev embedding, we know that (|u n | p−2 u n ) is a bounded sequence in L p p−1 (Ω, C). Moreover, by (3.12), we see that Consequently, by [27,Lemma 4.8], |u 0 | p−2 u 0 is the weak limit of the sequence (|u n | p−2 u n ) in L p p−1 (Ω, C). Hence, which means that I A (u 0 ) = 0, and so, u 0 is a weak solution of (P ). Now, let Ψ 1 n be the function defined as With the above notations, we are able to prove the following steps: Step 2.
Then, as in the previous step, Note that As Ψ 1 n = u n − u 0 in Ω, by Hölder inequality,

Recalling that by [1, Lemma 3] there holds
it follows that (1). As u 0 is critical point of I A , we have I A (u 0 ) = 0 and thus (3.19) holds.
If Ψ 1 n → 0 in H 1 A (Ω, C) the statements of the main result are verified. Thus, we can suppose that By using the fact that and I 0 (Ψ 1 n ) = o n (1), we have Therefore, By (3.22), there is α > 0 such that Denote by (y 1 n ) the center of a hypercube Q i in which Ψ 1 n L p (Q i ) = d n . We claim that (y 1 n ) is unbounded sequence in R N . Arguing by contradiction, let us suppose that (y 1 n ) is bounded in R N . Then, there is R > 0 such that On the other hand, since Ψ 1 n 0 in H 1 0 (R N , C), the local compactness of the Sobolev embedding gives against (3.27). Therefore, the sequence (y 1 n ) is unbounded. Since Ψ 1 n (· + y 1 n ) 0 = Ψ 1 n 0 ∀n ∈ N, we deduce that (Ψ 1 n (· + y 1 n )) is a bounded sequence in H 1 0 (R N , C). Then, there is Step 4. u 1 is a nontrivial weak solution of (P 0 ). First, by (3.27), we derive that u 1 = 0, and by a straightforward computation I 0 (Ψ 1 n (· + y 1 n ))ϕ = o n (1), ∀ϕ ∈ C ∞ 0 (R N , C). Thus, passing to the limit as n → +∞, we find i.e., the function u 1 is a nontrivial weak solution of problem (P 0 ). We can repeat this process obtaining the sequences Alves et al. Vol. 89 (2021) with |y j n | → +∞, as n → +∞ and where each function u j is a nontrivial weak solution of problem (P 0 ). Now, an inductive argument ensures that and Since u j is a nontrivial solution of (P 0 ), it follows that Proof. The argument is standard. However, we give the details for the reader's convenience. Thanks to our hypotheses, one has I A (u n ) → c and I A (u n ) → 0 as n → +∞, with c < c 0 . Without loss of generality, we can suppose that (u n ) is bounded in H 1 A (Ω, C). Then, up to some subsequence, there exists which contradicts c < c 0 . Thereby, u n u 0 and u n 2 → u 0 2 , and this implies that u n → u 0 in H 1 A (Ω, C). Proof. Assume by contradiction that u n → u 0 in H 1 A (Ω, C). By Lemma 3.2, it follows that k ≥ 1. As I 0 (u j ) ≥ c 0 , we must have k = 1, and so, We claim that u 0 = 0, otherwise (1) and thus u 1 is a critical point of I 0 with I 0 (u 1 ) < c 0 + κ 1 . Hence, by Theorem 2.3, we must have I 0 (u 1 ) = c 0 , which is impossible because c > c 0 . From this, u 0 = 0 and Hence, the limit equality which gives an absurd. Thus, we must have u n → u 0 in H 1 A (Ω, C). This shows the desired result.

Technical Estimates
The main goal this section is to establish some technical estimates that we will use in the proof of Theorem 1.1.
We start by introducing the following operator ϕ is the positive ground state of (P ∞ ) satisfying (2.4) and ζ, ξ are given as in the proof of Theorem 3.1. A direct computation ensures that the functions φ y,ρ belong to H 1 A (Ω, C) and L p (Ω, C), respectively. From now on, we take t y,ρ > 0 such that ψ ρ (y) = t y,ρ φ y,ρ ∈ N A .  Proof. (i) Note that Consequently, In the same way, we can show that On the other hand, note that for each j ∈ {1, .., N }, We claim that Indeed, note that we can deduce that (4.3) holds.
As ψ ρ (y) = t y,ρ φ y,ρ ∈ N A , it follows that This combined with diamagnetic inequality (2.1) leads to as ρ → 0, uniformly in y, we can infer that On the other hand, we also know that the limits below are uniform in y ∈ R N . These facts imply that there exists C > 0 that depends on M , which is independent of y ∈ R N , such that Therefore, using the fact that ϕ satisfies (2.4) and that c 0 = c ∞ , we obtain where Γ := C 2 2 . We point out that, from the above calculations, Γ is bounded when M → 0. This information will be useful in the next section.
(ii) For each fixed ρ, let us consider an arbitrary sequence (y n ) ⊂ R N with |y n | → ∞ as n → +∞ and let t y n ,ρ > 0 such that t y n ,ρ φ y n ,ρ ∈ N A . As in the proof of Theorem 3.1, |ϕ(x − yn)| p dx and ty n ,ρ → 1. In light of the previous lemma, we can prove the corollary below.
Fixing > 0 such that if A ∞ < then 2M Γ ϕ 2 L 2 (Ω) < κ 1 , we must have that sup This ends the proof of the corollary.
Hereafter, let us fix ρ ∈ (0, ρ 0 ), such that Furthermore, we consider the barycenter function given by where χ ∈ C(R + , R) is a non-increasing real function such that
Proof. Since T 0 ⊂ N A and we have Now we are going to show that c 0 = λ 0 . Suppose by contradiction that c 0 = λ 0 . Then, there exists a minimizing sequence (u n ) ⊂ T 0 ⊂ H 1 A (Ω, C) such that I A (u n ) → c 0 and τ (u n ) = 0 ∀n ∈ N.
By the Ekeland variational principle [30,Theorem 2.4], we can find a sequence (w n ) ⊂ N A such that A well-known computation shows that (u n ) and (w n ) are bounded. Moreover, there exists a sequence (λ n ) ⊂ R such that where J A (u) = I A (u), u . Using standard arguments, we have that λ n → 0, and then (4.7) yields Assuming that w n w 0 in H 1 A (Ω, C), we have that I A (w 0 ) = 0, and so, w 0 is a solution for (P ). Hence, by Theorem 3.1, we cannot have w n → w 0 in H 1 A (Ω, C), because this convergence would imply in I A (w 0 ) = c 0 = c A . Thereby, by Lemma 3.2, As I A (w 0 ) ≥ 0, then k = 1, w 0 = 0 and u 1 is a nontrivial solution of (P 0 ) with I 0 (u 1 ) = c 0 . From Theorem 2.3, there are θ > 0 and a ∈ R N such that (4.8) Since w n w 0 = 0, we get and Ψ n (· + y 1 n ) 2 A = w n (· + y 1 n ) 2 A → u 1 2 0 , where (y 1 n ) must be a sequence satisfying |y 1 n | → ∞. Therefore, w n (· + y 1 n ) → u 1 in H 1 A (R N , C). Setting u = u 1 , y n = y 1 n and v n (x + y 1 n ) = w n (x + y 1 n ) − u 1 (x), we have v n (x) = w n (x) − u(x − y n ) and v n 2 A = w n (· + y n ) − u 2 A . Therefore, the strong convergence of (w n (· + y n )) to u 1 yields v n → 0 in H 1 0 (R N , C). Next, we consider the following sets where the vector a is given in (4.8). Using the fact that |y n | → +∞ as n → +∞, we claim that there is a ball It is easy to see that (4.9) holds for r * > 0 small enough, because ϕ is positive, radial, strictly decreasing with respect to |x| and On the other hand, for each r * > 0 fixed, there is n 0 such that showing that B(y n + a, r * ) ⊂ (R N ) + n , for n large enough. Hence, for n large enough, |ϕ(x − y n − a)| 2 , χ(|x|), x, y n > 0 ∀x ∈ (R N ) + n , B(y n + a, r * ) ⊂ (R N ) * n , and |x| > R for every x ∈ B(y n + a, r * ). Using this information, we find Recalling that for each x ∈ (R N ) − n , |x − y n − a| ≥ |x|, and using again the fact that ϕ is radial with relation to the origin and decreasing, it follows that This fact, combined with the limit ϕ(· − y n − a) → 0 as |y n | → +∞ Therefore, by the Cauchy-Schwarz inequality and (4.10), τ (u(x − y n )), y n + a |y n + a| = (4.11) Now, using the fact that w n → u 1 in H 1 0 (R N , C) together with the limit τ (w n ) = o n (1), we find that τ (u(x − y n )) = o n (1), (4.12) which contradicts (4.11), and so, Now we are ready to prove the assertion (i) of Lemma 4.3. As ψ ρ (y) = t y,ρ φ y,ρ ∈ N 0 , by Theorem 3.1, I A (ψ ρ (y)) > c A = c 0 , ∀y ∈ R N .