Abstract
In this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation
where \(N \ge 3\), \(\Omega \subset {\mathbb {R}}^N\) is an exterior domain, \(p\in (2, 2^*)\) with \(2^*=\frac{2N}{N-2}\), and \(A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N\) is a continuous vector potential verifying \(A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .\)
1 Introduction
In this paper we investigate the existence of solutions for the following magnetic semilinear Schrödinger equation
where \(N \ge 3\), \(p\in (2, 2^*)\), \(2^*:=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\Omega \subset {\mathbb {R}}^N\) is an exterior domain, i.e. \(\Omega \) is an unbounded domain with smooth boundary \(\partial \Omega \ne \emptyset \) such that \({\mathbb {R}}^N {\setminus } \Omega \) is bounded, and \(A\in C({\mathbb {R}}^N, {\mathbb {R}}^N)\) satisfies
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
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 \(\Omega \); see for instance [9, 23] and [25].
In [12], Benci and Cerami studied the existence of nontrivial solutions for the problem
in an exterior domain \(\Omega \) without assuming symmetry, with \(2 \le p < 2^*\) and \(\lambda >0\). 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 \(p=1+\frac{8}{N}\) for \(N=5, 6, 7\) or \(p<\frac{2(N-1)}{N-2}\) for \(N=3, 4\),
-
there exists \(\lambda _{*}>0\) such that, for every \(\lambda \in (0, \lambda _{*})\), (1.2) has at least one positive solution,
-
for every \(\lambda \) there exists a \(\rho =\rho (\lambda )\) such that if \({\mathbb {R}}^{N}{\setminus } \Omega \subset B(x_{0}, \rho )\), with \(x_{0}\in {\mathbb {R}}^{N}{\setminus } \Omega \), (1.2) has at least one positive solution.
Later, existence results were obtained for more general problems
where f is a continuous function satisfying
In [9], Bahri and Lions studied (1.3) with \(f(x,t)=b(x) |t|^{p-2}t\) where \(b(x)\rightarrow b >0\) as \(|x|\rightarrow +\infty \). Using topological arguments, they showed that (1.3) has a solution when \(\Omega \) is an arbitrary exterior domain, for all \(\lambda >0\). 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 \(\Omega ={\mathbb {R}}^{N} {\setminus } B(0, R)\) 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 \({\mathbb {R}}^N\) 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
where \({\mathcal {D}} \subset {\mathbb {R}}^N\), with \(N \ge 2\), is a smooth domain, the function \(\psi \) takes values in \({\mathbb {C}}\), h is the Planck constant, i is the imaginary unit and \(A:{\mathbb {R}}^N \rightarrow {\mathbb {R}}^N\) denotes a magnetic potential. For the interested reader in this subject, we cite the papers by Alves, Figueiredo and Furtado [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:
Theorem 1.1
Suppose that (A) holds. Then, there exist \(\rho _0>0\) and \(\epsilon >0\) such that if \({\mathbb {R}}^N {\setminus } \Omega \subset B(0, \rho )\), \(\rho < \rho _0\) and \(\Vert A\Vert _\infty \le \epsilon \), 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 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 \(q \in (2, 2^*)\), we define \(q'\) as the conjugate exponent of q, that is, \(q' := \frac{q}{q-1}.\)
-
The usual norm of the Lebesgue spaces \(L^t(\Omega )\) for \(t \in [1,\infty ]\), will be denoted by \(|\,.\,|_{t}\), and the norm of the Sobolev space \(H^1_0(\Omega )\), by \(\Vert \,.\,\Vert \);
-
C denotes (possibly different) any positive constant.
2 Preliminary Results and the Limit Problem
In what follows, we denote by \(H_A^1(\Omega , {\mathbb {C}})\) the Hilbert space obtained by the closure of \(C_0^{\infty }(\Omega ,{\mathbb {C}})\) under the scalar product
where \(\Re (w)\) denotes the real part of \(w \in {\mathbb {C}}\), \({\overline{w}}\) is its complex conjugate, \(\nabla _Au:=(D_1u,D_2u,...,D_Nu)\) where \(D_j:=-i\partial _j+A_j(x)\), for \(j=1,2,...,N.\) The norm induced by this inner product is given by
We also consider the Hilbert space \(H_A^1({\mathbb {R}}^N, {\mathbb {C}})\) defined as
endowed with the scalar product
Then we can define the norm
By [26, Proposition 2.1-(i)], we know that \(C_{c}^{\infty }({\mathbb {R}}^N, {\mathbb {C}})\) is dense in \(H_{A}^{1}({\mathbb {R}}^N, {\mathbb {C}})\). A direct computation shows that \(H_A^1(\Omega , {\mathbb {C}}) \subset H_{A}^{1}({\mathbb {R}}^N, {\mathbb {C}})\).
As proved in [26, Sect. 2], for any \(u \in H_A^1({\mathbb {R}}^N, {\mathbb {C}})\), there holds the diamagnetic inequality, namely
where
Hence, if \(u\in H_{A}^{1}({\mathbb {R}}^N, {\mathbb {C}})\), then \(|u|\in H^1({\mathbb {R}}^N,{\mathbb {R}})\). Furthermore, as a consequence of the diamagnetic inequality, we have that the embedding
is continuous for any \(s\in [2, 2^*]\).
2.1 Limit Problem
In this subsection, we consider the scalar limit problem associated with (P), namely
where \(p\in (2,2^*)\). The reader is invited to see that \(u=u_{1}+\imath u_{2}\), 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 \({\mathcal {N}}_0\) is the Nehari manifold defined as
it is well-known (see [30]) that
from where it follows that \(c_0\) is the least energy of \((P_0)\). We recall that \(u\in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\) 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 only if, \(v(x):=|u(x)|\in H^1({\mathbb {R}}^N)\) is a least energy solution of
Moreover, \((P_0)\) and \((P_\infty )\) have the same least energy.
Proof
The proof can be done as in [24, Lemma 2.5]. \(\square \)
Lemma 2.2
The following facts hold:
-
(1)
\({c}_{0} >0\) is the least energy of \((P_\infty )\);
-
(2)
\({\mathcal {N}}_0 \ne \emptyset \);
-
(3)
\({c}_0\) is attained, and the set
$$\begin{aligned} {\mathcal {R}}_0:=\{u\in {\mathcal {N}}_0: { I}_0(u) = {c}_0,\;\;u(0) = \Vert u\Vert _\infty \} \end{aligned}$$is compact in \(H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\);
-
(4)
There exists \(C, c>0\) such that
$$\begin{aligned} |u(x)| \le Ce^{-c|x|}\;\;\forall x\in {\mathbb {R}}^N, u\in {\mathcal {R}}_0. \end{aligned}$$
Proof
See [24, Lemma 2.6]. \(\square \)
Let \(I_\infty :H^{1}({\mathbb {R}}^N,{\mathbb {R}}) \rightarrow {\mathbb {R}}\) be the energy functional given by
Note that \(I_{\infty }\) is defined on \(H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})\) while \(I_{0}\) is defined on \(H^{1}({\mathbb {R}}^{N}, {\mathbb {C}})\). If \({c}_\infty \) denotes the mountain pass level of \(I_{\infty }\) and \({\mathcal {N}}_\infty \) is the Nehari manifold given by
then (see [30])
Let \(\varphi \in H^1({\mathbb {R}}^N,{\mathbb {R}})\) be a positive ground state solution of \((P_\infty )\), that is,
The function \(\varphi \) 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 \(c_0=c_\infty \), and so, \(\varphi \) 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.
Theorem 2.3
There is \(\kappa >0\) such that if \(w \in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}}) \) is a critical point of \(I_0\) with \(I_0(w) \le c_0 +\kappa \), then there are \(a \in {\mathbb {R}}^N\) and \(\theta \in {\mathbb {R}}\) such that \(w(x)=e^{i \theta }\varphi (x-a)\), for all \(x \in {\mathbb {R}}^N\). Hence, \(I_0(w)=c_0\).
3 A Compactness Result for Energy Functional
In this section, we study some compactness property of the energy functional \(I_A: H_A^1(\Omega , {\mathbb {C}}) \rightarrow {\mathbb {R}}\) associated with (P) given by
In the sequel, we denote by \(c_A\) the mountain pass level of I that satisfies the equality below
where \({\mathcal {N}}_A\) is the Nehari manifold of \(I_A\) given by
Theorem 3.1
The equality \(c_0 = c_A\) holds true. Hence, there is no \(u\in H_A^1(\Omega , {\mathbb {C}})\) such that
and so, problem (P) has no ground state solution.
Proof
By using the diamagnetic inequality (2.1),
Recalling that \(\varphi \) satisfies (2.4) and that \(c_0=c_\infty \), we have that
Let \((y_n)\subset \Omega \) be a sequence such that \(|y_n|\rightarrow +\infty \), and \(\rho \) be the smallest positive number satisfying
Furthermore, let us define \(\zeta \in C^\infty ({\mathbb {R}}^N, [0,1])\) by
where \(\xi :[0, +\infty )\rightarrow [0,1]\) is a smooth non-decreasing function such that
Now, we consider the sequence
and fix \(t_n>0\) such that \(t_n \psi _n\in {\mathcal {N}}_A\). By making the change of variable \(z = x-y_n\), we deduce
Since \(|y_n|\rightarrow +\infty \) as \(n\rightarrow +\infty \), it is easy to check that
As
the dominated convergence theorem ensures that
or equivalently,
Recalling that \(\zeta (x) = 0\) for \(x\in {\mathbb {R}}^N {\setminus } \Omega \), by the previous discussion,
On the other hand, for each \(j \in \{1,..,N\}\),
Since \(\zeta (x+y_n) \rightarrow 1\), \(A(x + y_n) \rightarrow 0\) as \(n\rightarrow +\infty \) and \(\partial _j \zeta (x+y_n) \rightarrow 0\), the dominated convergence theorem implies that
and so,
By the previous analysis together with the fact that \(\varphi \in {\mathcal {N}}_0\), using translation invariance it is not difficult to prove that \(t_n \rightarrow 1\). Thus, by definition of \(c_A\), (3.3) and (3.4), we get
that leads to
Now, suppose by contradiction that there is \(v_0\in H_A^1(\Omega , {\mathbb {C}})\) such that
By (2.1), (3.6), and recalling that \(c_0=c_\infty \), we deduce that the function \(w=|v_0| \in H_{0}^{1}({\mathbb {R}}^N, {\mathbb {R}})\) is a ground state solution of \((P_\infty )\), that is,
Since \(w \ge 0\) in \({\mathbb {R}}^N\) and \(w \not =0\), the strong maximum principle ensures that \(w(x)>0\) for all \(x \in {\mathbb {R}}^N\), which is impossible because \(v_0 = 0\) in \({\mathbb {R}}^N {\setminus } \Omega \). \(\square \)
3.1 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 \({I}_0: H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\rightarrow {\mathbb {R}}\) associated with \((P_0)\) defined as in (2.3). With the above notations, we are able to prove the following compactness result.
Lemma 3.2
Let \((u_n)\subset H_A^1(\Omega , {\mathbb {C}})\) be a sequence such that
Then, up to a subsequence, there exists a weak solution \(u^0\in H_A^1(\Omega , {\mathbb {C}})\) of (P) such that
or there are k functions \((u_{n}^{j})\subset H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\), \(1\le j \le k\) such that
where \(u^j\) are nontrivial weak solutions of \((P_0)\), for every \(1\le j \le k\). Furthermore
and
Proof
We proceed by steps.
Step 1. The sequence \((u_n)\) is bounded in \(H_A^1(\Omega , {\mathbb {C}})\).
By (3.7),
for any \(\psi \in H_A^1(\Omega , {\mathbb {C}})\) and
Choosing \(\psi = u_n\) in (3.8), we obtain
which combined with (3.9) and (3.10) gives
Therefore, \((u_n)\) is bounded in \(H_A^1(\Omega , {\mathbb {C}})\).
Consequently, up to a subsequence, there exists \(u^0\in H_A^1(\Omega , {\mathbb {C}})\) such that
We claim that \(u^0\) is a weak solution of (P). In fact, for an arbitrary function \(\psi \in H_A^1(\Omega , {\mathbb {C}})\), the limit \({I}'_A(u_n) \rightarrow 0\) in \((H_A^1(\Omega , {\mathbb {C}}))^*\) yields
that is,
Since \(u_n \rightharpoonup u^0\) in \(H_A^1(\Omega , {\mathbb {C}})\), it follows that
From the boundedness of \((u_n)\) in \(H_A^1(\Omega , {\mathbb {C}})\) and Sobolev embedding, we know that \((|u_n|^{p-2}u_n)\) is a bounded sequence in \(L^{\frac{p}{p-1}}(\Omega , {\mathbb {C}})\). 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 \(L^{\frac{p}{p-1}}(\Omega , {\mathbb {C}})\). Hence,
which means that \({I}'_A(u^0) = 0\), and so, \(u^0\) is a weak solution of (P).
Now, let \(\Psi _n^1\) be the function defined as
With the above notations, we are able to prove the following steps:
Step 2.
Note that by the Brezis-Lieb lemma [15],
Moreover,
Using condition (A), it is easy to prove that
Therefore,
Consequently,
Now, (3.16) follows from (3.17) and (3.18).
Step 3.
For each \(v\in H_A^1(\Omega , {\mathbb {C}}) \subset H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}}) \) with \(\Vert v\Vert \le 1\), one has
and
Then, as in the previous step,
for every \(v\in H_A^1(\Omega , {\mathbb {C}})\) with \(\Vert v\Vert \le 1\). Consequently,
Now, we are going to show that
Note that
As \(\Psi _n^1 = u_n-u^0\) in \(\Omega \), by Hölder inequality,
Recalling that by [1, Lemma 3] there holds
it follows that
As \(u^0\) is critical point of \(I_A\), we have \({I}'_A(u^0) = 0\) and thus (3.19) holds.
If \(\Psi _n^1 \rightarrow 0\) in \(H_A^1(\Omega , {\mathbb {C}})\) the statements of the main result are verified. Thus, we can suppose that
By using the fact that
and \({I}'_{0}(\Psi _{n}^{1}) = o_n(1)\), we have
Therefore,
By (3.22), there is \(\alpha >0\) such that
Now, let us decompose \({\mathbb {R}}^N\) into N-dimensional unit hypercubes \(Q_i\) whose vertices have integer coordinates and put
Arguing as in [12, Lemma 3.1], there is \(\gamma >0\) such that
Denote by \((y_n^1)\) the center of a hypercube \(Q_i\) in which \(\Vert \Psi _{n}^{1}\Vert _{L^p(Q_i)} = d_n\). We claim that \((y_n^1)\) is unbounded sequence in \({\mathbb {R}}^N\). Arguing by contradiction, let us suppose that \((y_n^1)\) is bounded in \({\mathbb {R}}^N\). Then, there is \(R>0\) such that
On the other hand, since \(\Psi _{n}^{1} \rightharpoonup 0\) in \(H_0^1({\mathbb {R}}^N, {\mathbb {C}})\), the local compactness of the Sobolev embedding gives
against (3.27). Therefore, the sequence \((y_n^1)\) is unbounded. Since
we deduce that \((\Psi _{n}^{1} (\cdot + y_n^1))\) is a bounded sequence in \(H_0^1({\mathbb {R}}^N, {\mathbb {C}})\). Then, there is \(u^1\in H_0^1({\mathbb {R}}^N, {\mathbb {C}})\) such that
and
Step 4. \(u^1\) is a nontrivial weak solution of \((P_0)\).
First, by (3.27), we derive that \(u^1\ne 0\), and by a straightforward computation
Thus, passing to the limit as \(n \rightarrow +\infty \), we find
Now, the density of \(C_{0}^{\infty }({\mathbb {R}}^N,{\mathbb {C}})\) in \(H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\) ensures that
i.e., the function \(u^1\) is a nontrivial weak solution of problem (\(P_0\)).
We can repeat this process obtaining the sequences
with
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
for every \(1\le j\le k\). Now, the rest of the proof follows as in [12, Lemma 3.1]. \(\square \)
Corollary 3.3
Let \((u_n)\) be as in Lemma 3.2 with \(c< c_0\). Then \((u_n)\) admits a strongly convergent subsequence. Hence, the functional \({I}_A\) verifies the \((PS)_c\) condition, for every \(c \in \left( 0,c_0 \right) .\)
Proof
The argument is standard. However, we give the details for the reader’s convenience. Thanks to our hypotheses, one has
with \(c<c_0\). Without loss of generality, we can suppose that \((u_n)\) is bounded in \(H_A^1(\Omega , {\mathbb {C}})\). Then, up to some subsequence, there exists \(u^0\in H_A^1(\Omega , {\mathbb {C}})\) such that
If \(u \not \rightarrow u^0\) in \(H_A^1(\Omega , {\mathbb {C}})\), by Lemma 3.2 we must have \(k \ge 1\). Hence,
which contradicts \(c<c_0\). Thereby,
and this implies that \(u_n \rightarrow u^0\) in \(H_A^1(\Omega , {\mathbb {C}})\). \(\square \)
Corollary 3.4
Assume that there exists \((u_n)\) for which all the assumptions of Lemma 3.2 hold. If
where \(\kappa _1 < \min \{\kappa ,c_0\}\), then \((u_n)\) admits a strongly convergent subsequence. Hence, the energy functional \(I_A\) satisfies the \((PS)_c\) condition, for every \(c \in \left( c_0, c_0+\kappa _1 \right) ,\) where \(\kappa \) is given by Theorem 2.3.
Proof
Assume by contradiction that \( u_n \not \rightarrow u^0\) in \(H_A^1(\Omega , {\mathbb {C}})\). By Lemma 3.2, it follows that \(k \ge 1\). As \(I_0(u^j) \ge c_0\), we must have \(k=1\), and so,
We claim that \(u^0 \not = 0\), otherwise
and thus \(u^1\) is a critical point of \(I_0\) with \(I_0(u^1)<c_0+\kappa _1\). Hence, by Theorem 2.3, we must have \(I_0(u^1)=c_0\), which is impossible because \(c>c_0\). From this, \(u^0 \not = 0\) and
Hence, the limit equality
yields that
which gives an absurd. Thus, we must have \(u_n \rightarrow u^0\) in \(H_A^{1}(\Omega , {\mathbb {C}})\). This shows the desired result. \(\square \)
4 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
where
\(\varphi \) is the positive ground state of \((P_{\infty })\) satisfying (2.4) and \(\zeta , \xi \) are given as in the proof of Theorem 3.1. A direct computation ensures that the functions \(\phi _{y, \rho }\) belong to \(H_{A}^{1}(\Omega , {\mathbb {C}})\) and \(L^p(\Omega , {\mathbb {C}})\), respectively. From now on, we take \(t_{y,\rho }>0\) such that \(\psi _{\rho }(y) = t_{y,\rho }\phi _{y,\rho }\in {\mathcal {N}}_{A}\).
Lemma 4.1
The following relations hold:
-
(i)
\(\displaystyle \limsup _{\rho \rightarrow 0}I_A(\psi _{\rho }(y)) \le c_0+M\Gamma \Vert \varphi \Vert ^2_{L^{2}(\Omega )}\) uniformly in y;
-
(ii)
\(I_A(\psi _\rho (y)) \rightarrow c_0\) as \(|y|\rightarrow +\infty \), for every \(\rho \),
where \(M:=\sum _{j=1}^{N}\Vert A_j\Vert _\infty ^2\) and \(\Gamma >0\) is a constant independent of y.
Proof
(i) Note that
from where it follows that
Consequently,
In the same way, we can show that
and
On the other hand, note that for each \(j \in \{1,..,N\}\),
We claim that
Indeed, note that
which yields
Since
we can deduce that (4.3) holds.
As \(\psi _{\rho }(y) = t_{y,\rho }\phi _{y,\rho } \in {\mathcal {N}}_A\), it follows that
This combined with diamagnetic inequality (2.1) leads to
Now, recalling that
as \(\rho \rightarrow 0\), uniformly in y, we can infer that
On the other hand, we also know that the limits below
are uniform in \(y\in {\mathbb {R}}^N\). These facts imply that there exists \(C>0\) that depends on M, which is independent of \(y\in {\mathbb {R}}^N\), such that
Indeed,
Therefore, using the fact that \(\varphi \) satisfies (2.4) and that \(c_0=c_\infty \), we obtain
where \(\Gamma :=\frac{C^2}{2}\). We point out that, from the above calculations, \(\Gamma \) is bounded when \(M \rightarrow 0\). This information will be useful in the next section.
(ii) For each fixed \(\rho \), let us consider an arbitrary sequence \((y_n)\subset {\mathbb {R}}^N\) with \(|y_n|\rightarrow \infty \) as \(n\rightarrow +\infty \) and let \(t_{y_n, \rho }>0\) such that \(t_{y_n, \rho }\phi _{y_n,\rho }\in {\mathcal {N}}_{A}\). As in the proof of Theorem 3.1,
From this,
\(\square \)
In light of the previous lemma, we can prove the corollary below.
Corollary 4.2
There is \(\rho _0>0\) and \(\epsilon >0\) such that if \(\Vert A\Vert _\infty <\epsilon \), then
Proof
By Lemma 4.1—Part (i), one has
where \(M=\sum _{j=1}^{N}\Vert A_j\Vert _\infty ^2\), for every \(y\in {\mathbb {R}}^N\). So, there is \(\rho _0>0\) small enough such that
Fixing \(\epsilon >0\) such that if \(\Vert A\Vert _\infty <\epsilon \) then \(2M\Gamma \Vert \varphi \Vert _{L^{2}(\Omega )}^{2}<\kappa _1\), we must have that
This ends the proof of the corollary. \(\square \)
Hereafter, let us fix \(\rho \in (0, \rho _0)\), such that
Furthermore, we consider the barycenter function given by
where \(\chi \in C({\mathbb {R}}^+, {\mathbb {R}})\) is a non-increasing real function such that
for some \(R>0\) such that \({\mathbb {R}}^N {\setminus } \Omega \subset B(0, R)\). By definition of \(\chi \),
Set
Lemma 4.3
If
then
and there exists \(R_0>0\), with \(R_0>\rho \) such that:
-
(i)
If \(y\in {\mathbb {R}}^N\) with \(|y|\ge R_0\), then
$$\begin{aligned} I_A(\psi _\rho (y)) \in \Big (c_0, \frac{c_0+\lambda _0}{2} \Big ). \end{aligned}$$ -
(ii)
If \(y\in {\mathbb {R}}^N\) with \(|y| = R_0\), then
$$\begin{aligned} \langle \tau (\psi _\rho (y)), y\rangle >0. \end{aligned}$$
Proof
Since \({\mathcal {T}}_0 \subset {\mathcal {N}}_A\) and
we have
Now we are going to show that \(c_0\ne \lambda _0\). Suppose by contradiction that \(c_0 = \lambda _0\). Then, there exists a minimizing sequence \((u_n)\subset {\mathcal {T}}_0 \subset H_A^1(\Omega , {\mathbb {C}})\) such that
By the Ekeland variational principle [30, Theorem 2.4], we can find a sequence \((w_n)\subset {\mathcal {N}}_A\) such that
A well-known computation shows that \((u_n)\) and \((w_n)\) are bounded. Moreover, there exists a sequence \((\lambda _n)\subset {\mathbb {R}}\) such that
where \(J_A(u) = \langle I'_A(u), u\rangle \). Using standard arguments, we have that \(\lambda _n \rightarrow 0\), and then (4.7) yields
Assuming that \(w_n \rightharpoonup w^0\) in \(H_A^1(\Omega , {\mathbb {C}})\), 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 \(w_n \rightarrow w^0\) in \(H_A^1(\Omega , {\mathbb {C}})\), because this convergence would imply in \(I_A(w^0)=c_0=c_A\). Thereby, by Lemma 3.2,
As \(I_A(w^0)\ge 0\), 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 \(\theta >0\) and \(a \in {\mathbb {R}}^N\) such that
Since \(w_n \rightharpoonup w^0=0\), we get
and
where \((y_n^1)\) must be a sequence satisfying \(|y_n^1|\rightarrow \infty \). Therefore,
Setting
we have
Therefore, the strong convergence of \((w_n(\cdot + y_n))\) to \(u^1\) yields \(v_n \rightarrow 0\) in \(H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\).
Next, we consider the following sets
where the vector a is given in (4.8). Using the fact that \(|y_n|\rightarrow +\infty \) as \(n\rightarrow +\infty \), we claim that there is a ball
such that
It is easy to see that (4.9) holds for \(r^*>0\) small enough, because \(\varphi \) 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
Hence, for n large enough,
and \(|x|>R\) for every \(x\in B(y_n+a, r_*)\). Using this information, we find
Recalling that for each \(x\in ({\mathbb {R}}^N)_n^-\),
and using again the fact that \(\varphi \) is radial with relation to the origin and decreasing, it follows that
This fact, combined with the limit
ensures that
Therefore, by the Cauchy-Schwarz inequality and (4.10),
Now, using the fact that \(w_n \rightarrow u^1\) in \(H_{0}^{1}({\mathbb {R}}^N, {\mathbb {C}})\) together with the limit \(\tau (w_n) = o_n(1)\), we find that
which contradicts (4.11), and so,
Now we are ready to prove the assertion (i) of Lemma 4.3. As \(\psi _\rho (y) = t_{y, \rho }\phi _{y,\rho }\in {\mathcal {N}}_0\), by Theorem 3.1,
By Lemma 4.1-part (ii), for each \(\rho \) fixed
Thereby, for a given \(\epsilon _1 \in (0, \frac{\lambda _0-c_0}{2})\), there is \(R_0>0\) such that
From this
Finally, let us show assertion (ii) of Lemma 4.3, by definition of \(\psi _\rho (y)\) and arguing as above with |y| large enough, we derive
As \(t_{y, \rho } \rightarrow 1\) as \(|y|\rightarrow +\infty \), we have for \(|y| = R_0\) large,
\(\square \)
5 Proof of Theorem 1.1
In the sequel, we consider the sets
and
Lemma 5.1
If \(B\in \Upsilon \), then \(B\cap {\mathcal {T}}_0 \ne \emptyset \).
Proof
We are going to show that, for every \(B\in \Upsilon \), there exists \(u\in B\) such that \(\tau (u) = 0\). Equivalently, we prove that: for every \(h\in {\mathcal {H}}\), there exists \({\tilde{y}}\in {\mathbb {R}}^N\) with \(|{\tilde{y}}|\le R_0\) such that
For any \(h\in {\mathcal {H}}\), we set the functions
and \({\mathcal {F}}:[0,1]\times {\overline{B}}(0, R_0)\rightarrow {\mathbb {R}}^N\) given by
We claim that \(0\not \in {\mathcal {F}}(t, \partial B(0, R_0))\). Indeed, for \(|y| = R_0\), by Lemma 4.3—Part (i) we have
Hence,
and
Now
- \(\circ \):
-
If \(t=0\), then \(\langle {\mathcal {F}}(0,y), y\rangle = |y|^2 = R_{0}^{2}>0\);
- \(\circ \):
-
If \(t=1\), then by Lemma 4.3—Part (ii) we have \(\langle {\mathcal {F}}(1,y), y\rangle = \langle \tau (\psi _\rho (y)), y \rangle >0\);
- \(\circ \):
-
If \(t\in (0,1)\), then \(\langle {\mathcal {F}}(t,y), y\rangle >0\), since the terms \(t, 1-t, \langle \tau (\psi _\rho (y)), y\rangle \) and \(|y|^2\) are positives.
Then, by using the homotopy-invariance of the Brouwer degree, one has
Recalling that
there exists \({\tilde{y}}\in B(0, R_0)\) such that \({\mathcal {J}}({\tilde{y}}) = 0\), that is,
This completes the proof of Lemma 5.1. \(\square \)
Now, let us define
and
for every \(\gamma \in {\mathbb {R}}\).
Proof of Theorem 1.1. We choose \(\rho = \rho _0\), where \(\rho _0\) is given in Corollary 4.2. We claim that the constant c defined in (5.3) is a critical value for \(I_A\), that is, \({\mathcal {K}}_c \ne \emptyset \). We start our analysis by noting that
First of all, we recall that by Lemma 5.1, \(B\cap {\mathcal {T}}_0 \ne \emptyset \) for every \(B\in \Upsilon \). Then, for each \(B\in \Upsilon \), there is \({\tilde{u}}\in B \cap {\mathcal {T}}_0\) such that
Thus
Since
it follows that
Now, taking \(h \equiv I\), we find
The last inequality, together with Corollary 4.2 and (5.6) leads to
which proves (5.4).
Suppose by contradiction that \({\mathcal {K}}_c = \emptyset \). Recall that
By Corollary 3.4 and the deformation lemma [30], there is a continuous map
and a positive number \(\varepsilon _0\) such that
-
(a)
\(L_{c+\varepsilon _0}{\setminus } L_{c-\varepsilon _0} \subset \subset L_{c_0 + \kappa _1} {\setminus } L_{\frac{\lambda _0+c_0}{2}}\),
-
(b)
\(\eta (t,u) = u,\;\;\forall u\in L_{c-\varepsilon _0} \cup \{{\mathcal {N}}_A {\setminus } L_{c+\varepsilon _0}\}\;\;\text{ and }\;\;\forall t\in [0,1]\),
-
(c)
\(\eta (1, L_{c + \frac{\varepsilon _0}{2}}) \subset L_{c - \frac{\varepsilon _0}{2}}\).
Fix \({\tilde{B}}\in \Upsilon \) such that
Since
it follows that
Now, by (c), one has
that is,
On the other hand, we notice that \(\eta (1, \cdot ) \in C({\mathcal {N}}_A, {\mathcal {N}}_A)\). Moreover, since \({\tilde{B}} \in \Upsilon \), there exists \(h\in {\mathcal {H}}\) such that \({\tilde{B}} = h(\Sigma )\). Consequently,
Since \(h\in {\mathcal {H}}\), it follows that
and
Taking into account that
by item (b), we easily have
Then \({\tilde{h}} \in {\mathcal {H}}\). Moreover
owing to \(\eta (1, {\tilde{B}}) = {\tilde{h}}(\Sigma )\). Therefore, exploiting the definition of c, we have
which contradicts (5.9). Thereby, \({\mathcal {K}}_c \ne \emptyset \) and c is a critical value of \(I_A\) on \({\mathcal {N}}_A\), namely there is at least one nontrivial weak solution of (P). Hence, Theorem 1.1 is proved.
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C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7 and C. T. Ledesma was partially supported by CONCYTEC, Peru, 379-2019-FONDECYT “ASPECTOS CUALITATIVOS DE ECUACIONES NO-LOCALES Y APLICACIONES.
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Alves, C.O., Ambrosio, V. & Ledesma, C.E.T. An Existence Result for a Class of Magnetic Problems in Exterior Domains. Milan J. Math. 89, 523–550 (2021). https://doi.org/10.1007/s00032-021-00340-z
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DOI: https://doi.org/10.1007/s00032-021-00340-z
Keywords
- Variational methods
- Semilinear elliptic equations
- Schrödinger equation
Mathematics Subject Classification
- 35A15
- 35J61
- 35J10